# Computable Poincaré–Friedrichs constants for the $L^p$ de Rham complex over convex domains and domains with shellable triangulations

Théophile Chaumont-Frelet, Martin Werner Licht, Martin Vohralík

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Théophile Chaumont-Frelet, Martin Werner Licht, Martin Vohralík. Computable Poincaré–Friedrichs constants for the $L^p$ de Rham complex over convex domains and domains with shellable triangulations. 2025. hal-05204325

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Computable Poincaré–Friedrichs constants for the $L^p$ de Rham complex over convex domains and domains with shellable triangulations

Théophile Chaumont-Frelet* Martin Werner Licht† Martin Vohralík‡

Abstract

We construct potentials for the exterior derivative, in particular, for the gradient, the curl, and the divergence operators, over domains with shellable triangulations. Notably, the class of shellable triangulations includes local patches (stars) in two or three dimensions. The operator norms of our potentials satisfy explicitly computable bounds that depend only on the geometry. We thus compute upper bounds for constants in Poincaré–Friedrichs inequalities and lower bounds for the eigenvalues of vector Laplacians. As an additional result with independent standing, we establish Poincaré– Friedrichs inequalities with computable constants for the $L^p$ de Rham complex over bounded convex domains, derived as explicit operator norms of regularized Poincaré and Bogovskiï potential operators. We express all our main results in the calculus of differential forms and treat the gradient, curl, and divergence operators as instances of the exterior derivative. Computational examples illustrate the theoretical findings.

Contents

<table><thead><tr><th></th><th></th></tr></thead><tbody><tr><td><strong>1 Introduction</strong></td><td><strong>2</strong></td></tr><tr><td>1.1 Conceptual overview</td><td>3</td></tr><tr><td>1.1.1 Potentials and Poincaré–Friedrichs inequalities for the gradient</td><td>3</td></tr><tr><td>1.1.2 Potentials and Poincaré–Friedrichs inequalities for the curl and divergence</td><td>4</td></tr><tr><td>1.1.3 Potentials and Poincaré–Friedrichs inequalities for the exterior derivative</td><td>4</td></tr><tr><td>1.2 Literature review</td><td>5</td></tr><tr><td>1.2.1 General results</td><td>5</td></tr><tr><td>1.2.2 Analytical constants over convex domains</td><td>5</td></tr><tr><td>1.2.3 Domains star-shaped with respect to a ball</td><td>6</td></tr><tr><td>1.2.4 Triangulated domains</td><td>6</td></tr><tr><td>1.3 Objectives and methodology</td><td>7</td></tr><tr><td>1.3.1 Convex domains</td><td>7</td></tr><tr><td>1.3.2 Potentials subject to partial boundary conditions</td><td>7</td></tr><tr><td>1.3.3 Main results</td><td>8</td></tr><tr><td>1.4 Basic notation</td><td>8</td></tr><tr><td>1.5 Organization of this manuscript</td><td>8</td></tr><tr><td><strong>2 Review of Poincaré and Poincaré–Friedrichs inequalities</strong></td><td><strong>9</strong></td></tr><tr><td>2.1 Relationship with Poincaré inequalities</td><td>9</td></tr><tr><td>2.2 Analytical constants in Poincaré–Friedrichs inequalities over bounded convex domains</td><td>10</td></tr><tr><td><strong>3 Basic notions of triangulations</strong></td><td><strong>11</strong></td></tr><tr><td>3.1 Shape measures and related quantities</td><td>12</td></tr></tbody></table>

<footnote>*Inria Univ. Lille and Laboratoire Paul Painlevé, 59655 Villeneuve-d'Ascq, France, theophile.chaumont@inria.fr</footnote>

<footnote>†Corresponding author. École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland, martin.licht@epfl.ch</footnote>

<footnote>‡Inria, 48 rue Barrault, 75647 Paris, France & CERMICS, Ecole des Ponts, 77455 Marne-la-Vallée, France, martin.vohralik@inria.fr</footnote>
<table>
  <tbody>
    <tr>
      <td>4</td>
      <td>Poincaré-Friedrichs inequalities over triangulated domains</td>
      <td>15</td>
    </tr>
    <tr>
      <td>5</td>
      <td>Review of vector calculus and exterior calculus</td>
      <td>20</td>
    </tr>
    <tr>
      <td>5.1</td>
      <td>Vector calculus</td>
      <td>20</td>
    </tr>
    <tr>
      <td>5.2</td>
      <td>Exterior calculus</td>
      <td>21</td>
    </tr>
    <tr>
      <td>5.3</td>
      <td>Smooth differential forms</td>
      <td>22</td>
    </tr>
    <tr>
      <td>5.4</td>
      <td>Sobolev spaces of differential forms</td>
      <td>23</td>
    </tr>
    <tr>
      <td>5.5</td>
      <td>Transformations by bi-Lipschitz mappings</td>
      <td>23</td>
    </tr>
    <tr>
      <td>5.6</td>
      <td>Some approximation properties</td>
      <td>24</td>
    </tr>
    <tr>
      <td>6</td>
      <td>Regularized potentials over bounded convex sets</td>
      <td>24</td>
    </tr>
    <tr>
      <td>6.1</td>
      <td>Regularized Poincaré and Bogovskiï operators</td>
      <td>25</td>
    </tr>
    <tr>
      <td>6.2</td>
      <td>An operator norm bound with respect to the Lebesgue norm: the Poincaré case</td>
      <td>25</td>
    </tr>
    <tr>
      <td>6.3</td>
      <td>An operator norm bound with respect to the Lebesgue norm: the Bogovskiï case</td>
      <td>26</td>
    </tr>
    <tr>
      <td>6.4</td>
      <td>Rewriting the potentials operators</td>
      <td>27</td>
    </tr>
    <tr>
      <td>6.5</td>
      <td>Interaction of potentials with the exterior derivative</td>
      <td>28</td>
    </tr>
    <tr>
      <td>6.6</td>
      <td>Operator norms as bounds for the Poincaré-Friedrichs constants</td>
      <td>31</td>
    </tr>
    <tr>
      <td>7</td>
      <td>Shellable triangulations of manifolds</td>
      <td>33</td>
    </tr>
    <tr>
      <td>7.1</td>
      <td>Triangulations of manifolds</td>
      <td>33</td>
    </tr>
    <tr>
      <td>7.2</td>
      <td>Shellable simplicial complexes</td>
      <td>36</td>
    </tr>
    <tr>
      <td>8</td>
      <td>Reflections and deformations on shellable stars</td>
      <td>38</td>
    </tr>
    <tr>
      <td>9</td>
      <td>Constructive estimates of Poincaré-Friedrichs constants</td>
      <td>42</td>
    </tr>
    <tr>
      <td>10</td>
      <td>Numerical examples</td>
      <td>46</td>
    </tr>
    <tr>
      <td>10.1</td>
      <td>Software and algorithms</td>
      <td>46</td>
    </tr>
    <tr>
      <td>10.2</td>
      <td>Estimates for partial boundary conditions</td>
      <td>46</td>
    </tr>
    <tr>
      <td>10.3</td>
      <td>Two-dimensional examples</td>
      <td>47</td>
    </tr>
    <tr>
      <td>10.4</td>
      <td>Three-dimensional examples</td>
      <td>48</td>
    </tr>
    <tr>
      <td>11</td>
      <td>Outlook</td>
      <td>49</td>
    </tr>
    <tr>
      <td>A</td>
      <td>Outline of an alternative estimate</td>
      <td>50</td>
    </tr>
  </tbody>
</table>

# 1 Introduction

Potentials for the differential operators of vector calculus and exterior calculus are of fundamental impor-
tance. The operator norms of these potentials are upper bounds for the *Poincaré-Friedrichs constants*,
which quantify the fundamental stability properties of numerous partial differential equations and enter
the stability and convergence theory of numerical methods. Upper bounds for the Poincaré-Friedrichs
constants also provide lower bounds for the eigenvalues of the associated Laplacians. However, while
potentials and Poincaré-Friedrichs constants for the gradient have been subject to extensive study, quan-
tifiable results regarding the curl and divergence operators, or more generally, the exterior derivative,
are largely unavailable.

This manuscript contributes to the theory of computable estimates for Poincaré-Friedrichs constants for the differential operators of vector calculus. How to construct potentials for the gradient over triangulated domains is well-documented in the literature. Here, we extend this construction to the curl and divergence operators, treating them as instances of the exterior derivative. However, to proceed in the general exterior derivative case, we restrict our efforts to so-called *shellable triangulations*. The class of shellable triangulations includes practically relevant triangulations: for example, local triangulations around simplices within a larger triangulation (the so-called local patches or stars) are shellable in dimensions two and three; see also Figure 1. Only contractible domains can ever admit a shellable triangulation, but having computable upper bounds for such domains is an important stepping stone toward more general situations. Additionally, we include a study of regularized Poincaré and Bogovskiï
operators that leads to new Poincaré–Friedrichs inequalities with computable constants for the whole $L^p$ de Rham complex over bounded convex domains.

## 1.1 Conceptual overview

We give a conceptual overview of the topic before we outline the known results in the literature and our contributions in more detail. Our conceptual point of reference is the Poincaré–Friedrichs inequality for the gradient of scalar functions, which has been subject to extensive research.

### 1.1.1 Potentials and Poincaré–Friedrichs inequalities for the gradient

For the purpose of illustration, we let $\Omega \subseteq \mathbb{R}^3$ be a bounded connected open set. We let $L^p(\Omega)$ denote the Lebesgue space over $\Omega$ with integrability exponent $1 \le p \le \infty$, and we write $W^{1,p}(\Omega)$ for the first-order Sobolev space over $\Omega$ with integrability exponent $p$.

We are interested in finding a constant $C_{\Omega,\text{grad},p} > 0$ such that the following holds: for every gradient vector-valued field $f \in \nabla W^{1,p}(\Omega)$ there exists a scalar potential $u \in W^{1,p}(\Omega)$ (meaning a preimage under the gradient) such that $\nabla u = f$ and

$$ \|u\|_{L^p(\Omega)} \le C_{\Omega,\text{grad},p} \|f\|_{L^p(\Omega)}. \quad (1) $$

This inequality is called *Poincaré–Friedrichs inequality* and the constant $C_{\Omega,\text{grad},p}$ is called the *Poincaré–Friedrichs constant* with exponent $p$. The question is therefore whether we can always find a gradient potential of sufficiently small Lebesgue norm. One possible choice is the norm-minimizing gradient potential

$$ \Phi_{\text{grad}}(f) := \underset{\substack{u \in W^{1,p}(\Omega) \\ \nabla u = f}}{\operatorname{argmin}} \|u\|_{L^p(\Omega)}, \quad f \in \nabla W^{1,p}(\Omega). \quad (2) $$

In fact, its operator norm is the best Poincaré–Friedrichs constant:

$$ C_{\Omega,\text{grad},p} := \max_{u \in W^{1,p}(\Omega) \setminus \mathbb{R}} \frac{\|\Phi_{\text{grad}}(\nabla u)\|_{L^p(\Omega)}}{\|\nabla u\|_{L^p(\Omega)}}. \quad (3) $$

In the present setting, where $\Omega$ is connected, the different possible gradient potentials only differ by constants. Thus, $\Phi_{\text{grad}}(f)$ solves a one-dimensional convex minimization problem, and the constant $C_{\Omega,\text{grad},p}$ is the best possible constant in the inequality

$$ \min_{c \in \mathbb{R}} \|u - c\|_{L^p(\Omega)} \le C_{\Omega,\text{grad},p} \|\nabla u\|_{L^p(\Omega)}, \quad \forall u \in W^{1,p}(\Omega). \quad (\text{PF}) $$

Finding the norm-minimizing potential and the optimal Poincaré–Friedrichs constant is generally challenging because $\Phi_{\text{grad}}$ is a nonlinear operator unless $p = 2$. For that reason, research has focused on linear potential operators. Indeed, any linear potential operator $\Phi : \nabla W^{1,p}(\Omega) \to W^{1,p}(\Omega)$ satisfying $\nabla \Phi(f) = f$ for any $f \in W^{1,p}(\Omega)$ satisfies

$$ \max_{u \in W^{1,p}(\Omega) \setminus \mathbb{R}} \frac{\|\Phi_{\text{grad}}(\nabla u)\|_{L^p(\Omega)}}{\|\nabla u\|_{L^p(\Omega)}} \le \max_{u \in W^{1,p}(\Omega) \setminus \mathbb{R}} \frac{\|\Phi(\nabla u)\|_{L^p(\Omega)}}{\|\nabla u\|_{L^p(\Omega)}}. \quad (4) $$

Thus, its operator norm serves as an upper bound for the Poincaré–Friedrichs constant $C_{\Omega,\text{grad},p}$. This is a general principle: upper bounds for Poincaré–Friedrichs constants are easily obtained from linear potentials. We highlight this perspective because it is our theoretical underpinning when generalizing the discussion to the curl and divergence operators below.

One straightforward example for such a bounded linear potential operator is the average-free potential. This potential operator $\Phi_{\emptyset}$ is defined by

$$ \Phi_{\emptyset}(\nabla u) := u - u_{\Omega}, \quad u \in W^{1,p}(\Omega), \quad (5) $$

where $u_{\Omega}$ denotes the average of $u$ over $\Omega$. Its operator norm is the Poincaré constant $C_{\Omega,\emptyset,p} > 0$, which satisfies the inequality

$$ \|u - u_{\Omega}\|_{L^p(\Omega)} \le C_{\Omega,\emptyset,p} \|\nabla u\|_{L^p(\Omega)}, \quad \forall u \in W^{1,p}(\Omega). \quad (P) $$
We emphasize that the average-free gradient potential (5) is generally not the norm-minimizing gradient potential unless $p = 2$. Hence, the optimal Poincaré constant in (P) generally differs from the optimal Poincaré-Friedrichs in (PF). Considerable research efforts have gone into determining the best possible values for these constants. Upper estimates for these constants also correspond to lower bounds for the spectra of Neumann-Laplacians over those domains. The relationship between the Poincaré constant and the Poincaré-Friedrichs constant will be further elaborated upon in later sections of this manuscript.

### 1.1.2 Potentials and Poincaré-Friedrichs inequalities for the curl and divergence

The study of potentials and Poincaré-Friedrichs inequalities for the curl or divergence operators in vector calculus is substantially different, with much fewer results available in the literature. Consider the spaces of vector fields

$$ W^p(\text{curl}, \Omega) := \{u \in L^p(\Omega)^3 : \text{curl } u \in L^p(\Omega)^3\}, \qquad (6) $$

$$ W^p(\text{div}, \Omega) := \{u \in L^p(\Omega)^3 : \text{div } u \in L^p(\Omega)\}. \qquad (7) $$

Their members are those vector-valued fields in Lebesgue spaces whose distributional curls and divergences, respectively, are in Lebesgue spaces. In contrast to the gradient, this only requires that certain sums of distributional partial derivatives are integrable, which means that these spaces are not classical Sobolev spaces of vector-valued fields. Our objective is to find bounded potential operators (i.e., right inverses) for the differential operators

$$ \text{curl} : W^p(\text{curl}, \Omega) \to L^p(\Omega)^3, \quad \text{div} : W^p(\text{div}, \Omega) \to L^p(\Omega). $$

We are interested in the natural analogs for the Poincaré-Friedrichs inequality of the gradient (PF), which for the curl and divergence are the inequalities

$$ \min_{v \in W^p(\text{curl}, \Omega)} \|v\|_{L^p(\Omega)} \le C_{\Omega, \text{curl}, p} \|f\|_{L^p(\Omega)}, \quad \forall f \in \text{curl} \, W^p(\text{curl}, \Omega), \qquad (8) $$

$$ \min_{\substack{v \in W^p(\text{div}, \Omega) \\ \text{div } v=f}} \|v\|_{L^p(\Omega)} \le C_{\Omega, \text{div}, p} \|f\|_{L^p(\Omega)}, \quad \forall f \in \text{div} \, W^p(\text{div}, \Omega). \qquad (9) $$

The fundamental difference to the gradient is that the curl and divergence have infinite-dimensional kernels. The kernel of the gradient is the one-dimensional space of constant functions, and it is thus trivially complemented for all $p$, with a canonical choice of projection. By contrast, the kernels of the curl and divergence operators are generally infinite-dimensional. Correspondingly, the curl and the divergence do not admit any easily available analog to the Poincaré inequality (P). Moreover, it is not immediately evident that the kernels of the curl and divergence operators are complemented in the Banach space case when $p \ne 2$, and a canonical projection only exists here in the Hilbert setting $p = 2$. In the light of these facts, there generally is no natural analog to the Poincaré inequality (P) for the curl and divergence.

In the Banach space case, not even the existence of norm-minimizing potentials is trivial. We are therefore interested in any potentials

$$ \Phi_{\text{curl}} : \text{curl} \, W^p(\text{curl}, \Omega) \to W^p(\text{curl}, \Omega), \qquad (10) $$

$$ \Phi_{\text{div}} : \text{div} \, W^p(\text{div}, \Omega) \to W^p(\text{div}, \Omega), \qquad (11) $$

and their operator norms. We remark that upper bounds for the Poincaré-Friedrichs inequality of the curl operator correspond to lower bounds for the so-called Maxwell eigenvalues. In sharp contrast to the extensive research on gradient potentials, not many results seems available on computable constants in such Poincaré-Friedrichs inequalities for the curl and divergence operators.

### 1.1.3 Potentials and Poincaré-Friedrichs inequalities for the exterior derivative

Though the above discussion is presented in vector calculus, our main results and arguments are given in the formalism of exterior calculus. Exterior calculus [37, 48] is used ubiquitously in the mathematical literature of physics and engineering and has found widespread adoption in the theoretical and numerical analysis for vector-valued partial differential equations [41, 38, 4, 5, 6, 23, 33, 7]. This formalism is
independent of the spatial dimension and highlights the underlying geometric structures common to the gradient, curl, and divergence operators in three space dimensions. In this setting, the Poincaré–Friedrichs inequalities we are interested in read as follows: For every $u \in W^p \Lambda^k(\Omega)$, there exists $w \in W^p \Lambda^k(\Omega)$ with the same exterior derivative $dw = du$ and such that<fnref n="1" />

$$ \|w\|_{L^p(\Omega)} \leq C_{\Omega,k,p} \|dw\|_{L^p(\Omega)}. \qquad (12) $$

For the purpose of our discussion, this formalism allows us to leverage results from a larger body of literature in differential geometry and functional analysis.

## 1.2 Literature review

We review the literature on Poincaré–Friedrichs inequalities. We also identify the obstructions inherent to presently known results that we wish to overcome with the present contribution.

### 1.2.1 General results

The qualitative existence of Poincaré–Friedrichs inequalities for the differential operators of vector calculus and exterior calculus is known for a large class of domains. We emphasize that non-constructive arguments, such as the Rellich embedding theorem, generally do not yield explicitly computable constants.

There is a vast body of literature on the Poincaré inequality for the gradient and its numerous variants, which may include weighted integrals or boundary terms. The divergence operator has received less attention. In the Hilbert space case $p = 2$, the Friedrichs inequality [15] over the Sobolev space $W^{1,2}(\Omega)$ with homogeneous Dirichlet boundary conditions along $\partial\Omega$ yields the Poincaré–Friedrichs constant for the divergence operator in (9) by duality. However, that easy duality argument, which yields an explicit upper bound proportional to the domain diameter, seems inherently restricted to $p = 2$.

The core challenges are found in the discussion of the curl operator in three dimensions, where Poincaré–Friedrichs inequalities have appeared under different names. Let us assume momentarily that $\Omega$ is a weakly Lipschitz domain with trivial topology and consider only the Hilbert case $p = 2$. Then the constant in (8) agrees with the constant in the so-called Poincaré–Friedrichs–Weber inequality

$$ \|u\|_{L^2(\Omega)} \leq C_{\Omega, \mathrm{curl}, p} \|\mathrm{curl} \, u\|_{L^2(\Omega)}, \qquad (13) $$

valid for all $u \in W^2(\mathrm{curl}, \Omega) \cap W^2(\mathrm{div}, \Omega)$ that satisfy $\mathrm{div} \, u = 0$ and have vanishing normal or vanishing tangential trace along $\partial\Omega$. Equivalently, (13) is valid for all $u \in W^2(\mathrm{curl}, \Omega)$ that are $L^2$-orthogonal to the gradients of scalar-valued fields in $W^{1,2}(\Omega)$ or that have vanishing tangential trace and are $L^2$-orthogonal to the gradients of those scalar-valued fields in $W^{1,2}(\Omega)$ that satisfy Dirichlet boundary conditions. We refer to Equation (5) in [30], Equation (2) in [31], [69] as well as [34, Lemmas 3.4 and 3.6], [28, Proposition 7.4], [59] and the references therein. The general case of $L^p$ differential forms over Lipschitz manifolds subject to partial boundary conditions is discussed in [36]. However, while many of the above results rely on non-constructive estimates, we are interested in practically computable upper bounds.

### 1.2.2 Analytical constants over convex domains

Considerable research effort has gone into computing explicit upper estimates for the constants in the Poincaré–Friedrichs inequalities over convex domains. Notably, if the constants are required to depend on the convex domain only via its diameter, then the optimal gradient Poincaré–Friedrichs and Poincaré constants for the entire range of Lebesgue exponent $1 \leq p < \infty$ are known explicitly [60, 9, 1, 26, 29], see also [21].

The literature on Poincaré–Friedrichs constants and potentials of curls and divergences is less extensive than for potentials of gradients, even over convex domains. Guerini and Savo [39] address the spectrum of the Hodge–Laplace operator on bounded convex domains with smooth boundary in the Hilbert space case $p = 2$. Among their results is the observation that the Poincaré–Friedrichs constant for the gradient already estimates the corresponding constants for the curl and divergence operators.

<footnote n="1">¹All the notation is fixed in detail in Section 5 later.</footnote>
They also provide explicit (but not necessarily optimal) upper bounds for Poincaré-Friedrichs constants that depend only on the dimension and diameter of the convex domain. A duality argument then yields upper estimates of the Poincaré-Friedrichs constants for the gradient, curl, and divergence operators subject to Dirichlet, tangential, and normal boundary conditions, respectively, along the entire boundary. However, no results as those of [39] are known over bounded convex Lipschitz domains and for general Lebesgue exponents $1 \le p \le \infty$.

### 1.2.3 Domains star-shaped with respect to a ball

When the domain is not necessarily convex but star-shaped with respect to a ball, then several estimates for Poincaré-Friedrichs constants are known.<fnref n="2" /> Polynomial interpolation estimates already imply the gradient Poincaré-Friedrichs inequality [13, 24]. We pay particular attention to the regularized Poincaré and Bogovskiï potential operators for the exterior derivative, such as those of Costabel and McIntosh [22]. If we conceive these potentials as mappings between Lebesgue spaces of differential forms, then their operator norms are upper estimates for the Poincaré-Friedrichs constants of the domain. We are aware of estimates for the higher-order seminorms of these potentials [40], but estimates in Lebesgue norms have not been made explicit in the literature yet, to the best of our knowledge. We particularly emphasize that all these estimates for domains star-shaped with respect to a ball have in common that they rely on upper bounds for the *eccentricity* of the domain.

Let us briefly discuss the practical limitations of the aforementioned estimates for Poincaré-Friedrichs constants for bounded convex domains or domains that are star-shaped with respect to a ball. Recall that the main objective of this manuscript is bounding Poincaré-Friedrichs constants over domains with shellable triangulations, with a key application being local stars within triangulated domains. Not all local stars describe convex subdomains. Even if the local stars are star-shaped with respect to a ball, which would enable the averaged Poincaré and Bogovskiï operators [22], the estimates that rely on this geometric condition deteriorate when the aforementioned ball has a radius much smaller than the domain diameter. While this is not as much a problem over local patches (stars) around interior subsimplices, where the size of the interior ball only depends on the shape regularity of the triangulation, the interior ball can be arbitrarily small when the local patch is around a boundary simplex, even if the triangulation displays decent shape regularity. This occurs most prominently when the domain has sharp reentrant corners, illustrative limit cases including the slit domain [66] and the crossed bricks domain [50], which contain local finite element patches that are not star-shaped with respect to any ball. In view of this, we refrain from treating local patches (stars) in triangulations as domains star-shaped with respect to a ball.

### 1.2.4 Triangulated domains

Geometric settings that admit finite triangulations enable different pathways to estimate the constant in Poincaré-Friedrichs inequalities. We review the main outcomes.

Computable estimates for Laplacian eigenvalues over triangulated domains have received much attention. The constant in (1) for $p = 2$ corresponds to a lower bound for the Laplace eigenvalues and quantifies the stability properties of the Laplacian on the domain $\Omega$. Similarly, the constant in (8) for $p = 2$ corresponds to a lower bound for the Maxwell eigenvalues and quantifies the stability properties of the Maxwell system on $\Omega$. Thus, computable upper bounds on the Poincaré-Friedrichs constants also give computable lower bounds for the eigenvalues of the associated Laplacians and vice versa. Prominent methods numerically compute guaranteed upper bounds on the Poincaré-Friedrichs constants upon solving a finite element system over a sufficiently fine triangulation and using some clever post-processing estimates. This approach has led to estimates for scalar Laplacian eigenvalues [17, 52, 16] and vector Laplacian eigenvalues [32]. For the purposes of this manuscript, however, we aim for computable upper bounds that do not rely on the solution of (global) finite element systems.

There are numerous estimates for Poincaré-Friedrichs constants that only rely on locally computable geometric quantities, such as the diameter and volumes of simplices. Veeser and Verfürth [66] provide computable upper bounds in the case of the classical Sobolev space $W^{1,p}(\Omega)$ over vertex stars. Naturally, their estimates depend on the shape regularity of the mesh. A whole class of upper bounds for

<footnote n="2">²In fact, estimates for Poincaré-Friedrichs constants even hold over any star-shaped open bounded set; see [42, Theorem 3.1].</footnote>
Poincaré-Friedrichs inequalities uses some form of passing through the triangulation and constructs the potentials step-by-step. The underlying idea is that we first construct a potential for the gradient over an initial simplex. Every time we have found a potential over a subdomain, we construct a potential over a neighboring simplex or patch: along the interfacing intersection, the two potentials will only differ by a constant, and that difference can easily be removed to ensure continuity across that interface. Cell by cell, the potential is constructed over increasing subdomains, matching along the interfacing intersections until the entire domain is exhausted. The method is known in the finite element literature [24]. It was previously used in the context of finite volume methods [27], broken (weakly continuous) Sobolev spaces [67], or more recently in continuous-discrete comparison results [12, 25, 19, 68]. This sequential procedure applies to general triangulated domains, not only local stars. Most importantly, these sequential estimates of Poincaré-Friedrichs constants generally circumvent the effect of low boundary regularity and only rely on the mesh's shape regularity.

While we thus know Poincaré-Friedrichs constants over local stars for scalar-valued Sobolev spaces, we are not aware of computable estimates for the case of $W^p(\text{curl}, \Omega)$ and $W^p(\text{div}, \Omega)$ over finite element stars.

## 1.3 Objectives and methodology

The main objective of this manuscript is the construction of potentials for the gradient, curl, and divergence operators, and, more generally, the exterior derivative. The operator norms of our potentials satisfy computable upper bounds, thus yielding computable upper estimates of the Poincaré-Friedrichs constants as well. In what follows, we focus on domains with shellable triangulations, which include local patches (stars) in two and three dimensions as important special cases. We also devote an important effort specifically to convex domains.

### 1.3.1 Convex domains

Our main result for convex domains in the exterior calculus setting is the construction of regularized Poincaré and Bogovskiǐ potential operators with explicitly bounded operator norms, as summarized in Theorem 6.2. These upper bounds for the Poincaré-Friedrichs constants are proportional to the domain diameter and are bounded in terms of the domain's eccentricity. The bounds are independent of the Lebesgue exponent $1 \le p \le \infty$, though the space dimension and the form degree enter the estimates.

The reason for our study of Poincaré-Friedrichs constants over convex domains is twofold: firstly, they are of independent interest, and secondly, we will need them as a component for our main results on triangulations. Our exposition of regularized Poincaré and Bogovskiǐ-type potential operators follows the general methodology of Costabel and McIntosh [22]. By comparison, we simplify the potential operators: we only study them over convex domains instead of domains star-shaped with respect to a ball. Moreover, we use simpler (constant) weight functions. While the resulting potentials feature generally lower regularity, they are conducive to our purposes. Crucially, this allows us to estimate their operator norms easily and thus bound Poincaré-Friedrichs constants.

### 1.3.2 Potentials subject to partial boundary conditions

We also address the construction of potentials for the gradient, curl, and divergence operators, being special cases of the exterior derivative, over a simplex subject to partial boundary conditions. We are not aware of explicit estimates or regularized potentials for these boundary conditions in the published literature. For example, given a divergence-free vector-valued field over a tetrahedron with vanishing normal trace along three of the tetrahedron's faces, we want to find the vector-valued field potential that not only is a preimage under the curl operator but also has a vanishing tangential trace along the same three faces. We achieve this by constructing an auxiliary problem subject to full boundary conditions, building upon the regularized Bogovskiǐ operators. We thus obtain Poincaré-Friedrichs inequalities over simplices and subject to partial boundary conditions. Again, we address the entire range $1 \le p \le \infty$ of Lebesgue exponents, and our Poincaré-Friedrichs constants are explicitly computable.
### 1.3.3 Main results

Our main objective remains to find potentials for the differential operators of vector calculus and exterior calculus, including the curl and divergence, over a broad class of triangulated domains. The operator norms of these potentials will serve as our computable Poincaré-Friedrichs constants. In light of the different approaches discussed above, we seek constants that are explicitly computed in terms of mesh geometry, that do not require the solution of global finite element problems, and that do not depend on the boundary regularity of the domain.

As discussed earlier, the sequential construction of potentials for gradient vector-valued fields is well-established in the literature and serves as our conceptual blueprint. Gradient potentials are easily computed over each individual simplex, but the constants of integration generally do not match, so the scalar piecewise potential will belong to a broken Sobolev space. The local potentials will differ by a constant along the simplex boundaries. However, we can sequentially construct a potential in Sobolev spaces over increasingly larger intermediate subdomains: at each step, we select a simplex that shares at least one face with one of the previously processed simplices and adjust the constant integration of the local potential. The global potential is built cell by cell until the entire domain is covered. Our main results in this context are Theorem 4.4 and Theorem 4.5.

We generalize this sequential construction of potentials to the curl, the divergence, and, more generally, the exterior derivative. However, we need to overcome new challenges that arise due to the infinite-dimensional kernels of these differential operators, as we now explain in more detail. The basic inductive strategy remains the same. We start by constructing, for instance, a curl potential over a single simplex. Having already defined a potential operator over a subtriangulation, we select a neighboring simplex that shares at least one face with the preceding simplices. We then construct a potential for the curl operator whose tangential traces along the shared faces match those of the already existing potential. Repeating this procedure eventually exhausts the original triangulation. Our main results here are Theorem 9.3 and Theorem 9.4.

However, unlike in the potential construction of the gradient, it is not immediately evident whether constructing the local curl potential with given tangential traces is a well-posed auxiliary problem. For the case of the gradient, it is sufficient that the sequential traversal of the triangulation satisfies that each new simplex shares at least one face with one of the previous simplices. For the differential operators of vector calculus and exterior calculus, it seems that we must be more restrictive: we require the new simplex to intersect with the existing subtriangulation along a boundary submanifold of dimension $n-1$, which means a collection of faces of this simplex but no individual edges or vertices, allowing us to define a well-posed auxiliary problem and to extend the existing curl potential to the new simplex. Whether a triangulation admits such a particular traversal is a non-trivial condition and defines the class of *shellable* triangulations.

Shellable simplicial complexes, and more generally polytopal complexes, are a well-established notion in discrete geometry and combinatorics; see, e.g., Kozlov [43] and Ziegler [71], and the references therein. As already highlighted, local patches (stars) in 2D and 3D triangulations are shellable. Generally speaking, domains that have a shellable triangulation must be contractible. Not all triangulations of contractible domains are shellable, but the exceptions are rare and requiring the triangulation to be shellable poses little restriction in practice.

## 1.4 Basic notation

Whenever $x \in \mathbb{R}^n$ is a vector, we write $\|x\| = \|x\|_2$ for its Euclidean norm, and whenever $A \in \mathbb{R}^{n \times n}$, we let $\|A\|_2$ be its operator norm with respect to the Euclidean norm. Furthermore, $\mathbf{J}F$ always denotes the Jacobian of any mapping $F$.

## 1.5 Organization of this manuscript

The remainder of this manuscript is structured as follows. We review Poincaré-Friedrichs constants for the gradient over convex domains in Section 2, where we also discuss the difference with the Poincaré inequality and motivate our interest in linear potentials. We review basic notions of triangulations in Section 3. Subsequently, we develop the computable upper bounds for the Poincaré-Friedrichs constants for the gradient over face-connected triangulated domains in Section 4. Sobolev spaces in vector calculus
and the calculus of differential forms are reviewed in Section 5. We then introduce our regularized potentials over convex sets in Section 6, giving rise to computable Poincaré–Friedrichs constants as their operator norms. We subsequently review shellable triangulations of manifolds in Section 7, and we construct an important geometric reflection operator in Section 8. Finally, we provide computable upper bounds for Poincaré–Friedrichs constants for the exterior derivative over shellable triangulations in Section 9 and present numerical examples in Section 10. We conclude with an outlook in Section 11.

## 2 Review of Poincaré and Poincaré–Friedrichs inequalities

This section surveys variations of the Poincaré–Friedrichs inequalities for the gradient operator, with emphasis on analytical upper bounds over bounded convex domains. Continuing the discussion from the introduction (Section 1.1.1), we detail the difference between the Poincaré–Friedrichs inequality, which addresses the norm-minimizing potential, and the Poincaré inequality, which addresses the potential with mean value zero. This survey serves as a building block in constructing computable constants over triangulated domains in a combinatorial way below (Section 4), but we believe it is also of independent interest.

Let $\Omega \subseteq \mathbb{R}^n$ be a connected open set. Given any $p \in [1, \infty]$, we let $L^p(\Omega)$ denote the Lebesgue space over $\Omega$ with integrability exponent $p$, and we write $L^p(\Omega) := L^p(\Omega)^n$ for the corresponding Lebesgue space of vector-valued fields. We also write $W^{1,p}(\Omega)$ for the first-order Sobolev space over $\Omega$ with integrability exponent $p$.

We say that a domain $\Omega \subseteq \mathbb{R}^n$ satisfies the *Poincaré–Friedrichs inequality* with exponent $p \in [1, \infty]$ if there exists a constant $C_{\Omega, \text{grad}, p} \ge 0$ such that the following holds: for every vector-valued field $f \in \nabla W^{1,p}(\Omega)$ there exists $u \in W^{1,p}(\Omega)$ such that $\nabla u = f$ and

$$ \|u\|_{L^p(\Omega)} \le C_{\Omega, \text{grad}, p} \|f\|_{L^p(\Omega)}. \quad (14) $$

Since $\Omega$ is connected, this is equivalent to

$$ \min_{c \in \mathbb{R}} \|u - c\|_{L^p(\Omega)} \le C_{\Omega, \text{grad}, p} \|\nabla u\|_{L^p(\Omega)}, \quad \forall u \in W^{1,p}(\Omega). $$

We call $C_{\Omega, \text{grad}, p}$ the *Poincaré–Friedrichs constant* with exponent $p$.

### 2.1 Relationship with Poincaré inequalities

We wish to clarify the relationship between the Poincaré–Friedrichs inequality, in the sense introduced above, with other inequalities that are known as Poincaré inequality (or also Poincaré–Wirtinger or Friedrichs inequality) in the literature [24, Remark 3.32]. Given $p \in [1, \infty]$ and a domain $\Omega \subseteq \mathbb{R}^n$ of finite measure, we say that $\Omega$ satisfies the Poincaré inequality with exponent $p$ if there exists $C_{\emptyset, p} \ge 0$ such that

$$ \|u - u_{\Omega}\|_{L^p(\Omega)} \le C_{\Omega, \emptyset, p} \|\nabla u\|_{L^p(\Omega)}, \quad \forall u \in W^{1,p}(\Omega), $$

where $u_{\Omega}$ is the *average* of $u$ over $\Omega$, that is,

$$ u_{\Omega} := \text{vol}(\Omega)^{-1} \int_{\Omega} u(x) \, dx. $$

Clearly, this Poincaré inequality implies the Poincaré–Friedrichs inequality and we have

$$ C_{\Omega, \text{grad}, p} \le C_{\Omega, \emptyset, p}. $$

Towards a converse inequality, let us first observe that the average of any $u \in W^{1,p}(\Omega)$ with $p < \infty$ satisfies the bound

$$ \|u_{\Omega}\|_{L^p(\Omega)}^p = \int_{\Omega} \left( \text{vol}(\Omega)^{-1} \int_{\Omega} |u(x)| \, dx \right)^p \le \int_{\Omega} \text{vol}(\Omega)^{-1} \int_{\Omega} |u(x)|^p \, dx = \|u\|_{L^p(\Omega)}^p. \quad (15) $$
Here, we have used Hölder's or Jensen's inequality. In the case $p = \infty$, any $u \in L^\infty(\Omega)$ satisfies $\|u_\Omega\|_{L^\infty(\Omega)} \le \|u\|_{L^\infty(\Omega)}$. We conclude that taking the average is a projection within Lebesgue spaces with unit norm. The triangle inequality now shows that

$$ \|u - u_\Omega\|_{L^p(\Omega)} \le 2\|u\|_{L^p(\Omega)}, \quad \forall u \in L^p(\Omega). $$

Thus, the Poincaré-Friedrichs inequality implies the Poincaré inequality with

$$ C_{\Omega, \mathrm{grad}, p} \le 2C_{\Omega, \emptyset, p}. \qquad (16) $$

In the special case $p = 2$, taking the average is an orthogonal projection, and so this improves to $\|u - u_\Omega\|_{L^2(\Omega)} \le \|u\|_{L^2(\Omega)}$ for any $u \in L^2(\Omega)$. Hence,

$$ C_{\Omega, \emptyset, 2} = C_{\Omega, \mathrm{grad}, 2}. \qquad (17) $$

This improvement also follows from the projection estimate (see, e.g., [70]). Stern's generalized projection estimate [65, Theorem 4.1, Remark 5.1] implies improved estimate for all $L^p$ spaces with $1 \le p \le \infty$: since taking the average is a projection onto the constants functions with unit norm, from (15) it now follows that

$$ \|u - u_\Omega\|_{L^p(\Omega)} \le \min(2, 2^{2/p-1}) \|u\|_{L^p(\Omega)} = 2^{2/p-1} \|u\|_{L^p(\Omega)}, \quad \forall u \in L^p(\Omega). $$

Here, we have used $1 \le 2^{2/p-1} \le 2$ for $1 \le p \le \infty$. We thus conclude

$$ C_{\Omega, \mathrm{grad}, p} \le 2^{2/p-1} C_{\Omega, \emptyset, p}. \qquad (18) $$

In the limit cases $p = 1$ and $p = \infty$ we reproduce (16), and in the case $p = 2$ we achieve the identity (17) once again. In summary, our notion of Poincaré-Friedrichs constant is equivalent to the common notion of Poincaré constant, up to a numerical factor that only depends on $1 \le p \le \infty$ and that is at most 2.

As discussed in Section 1, the notion of Poincaré-Friedrichs inequality suits our discussion better than the Poincaré inequality. The kernel of the gradient is the one-dimensional space of constant functions, and is thus complemented in the Lebesgue spaces with a canonical choice of projection. By contrast, the curl and divergence operators have infinite-dimensional kernels, and so it is not even trivial whether these kernels are complemented subspaces and admit a projection onto them, not to mention a canonical projection.

## 2.2 Analytical constants in Poincaré-Friedrichs inequalities over bounded convex domains

We collect examples for Poincaré and Poincaré-Friedrichs inequalities for the important special case of bounded convex domains. We have the Poincaré inequalities [60, 9, 1] (or [24, Lemma 3.24])

$$ \|u - u_\Omega\|_{L^1(\Omega)} \le \frac{\delta(\Omega)}{2} \|\nabla u\|_{L^1(\Omega)}, \quad \forall u \in W^{1,1}(\Omega), \qquad (19) $$

$$ \|u - u_\Omega\|_{L^2(\Omega)} \le \frac{\delta(\Omega)}{\pi} \|\nabla u\|_{L^2(\Omega)}, \quad \forall u \in W^{1,2}(\Omega), \qquad (20) $$

where $\delta(\Omega)$ is the diameter of the domain $\Omega$. These two estimates are the best possible Poincaré inequalities in the cases $p = 1$ and $p = 2$, respectively, in terms of the diameter alone. Upper bounds for the Poincaré constant over bounded convex domains with $1 < p < \infty$ are known in the literature [21, Theorem 1.1, Theorem 1.2]:

$$ \|u - u_\Omega\|_{L^p(\Omega)} \le C_{\mathrm{CW},p} \delta(\Omega) \|\nabla u\|_{L^p(\Omega)}, \quad \forall u \in W^{1,p}(\Omega), \qquad (21) $$

where we use an upper bound by Chua and Wheeden:

$$ C_{\mathrm{CW},p} := \sup_{v \in C^\infty([0,1]) \setminus \mathbb{R}} \frac{\|v - v_{[0,1]}\|_{L^p([0,1])}}{\|\nabla v\|_{L^p([0,1])}} \le \sqrt[p]{p} 2^{1-\frac{1}{p}} = 2 \left(\frac{p}{2}\right)^{\frac{1}{p}}. \qquad (22) $$
Note that (21) is generally not optimal among the upper bounds that only depend on the domain diameter and the Lebesgue exponent. As discussed above, these Poincaré inequalities imply Poincaré-Friedrichs inequalities.

We know optimal Poincaré-Friedrichs constants over bounded convex domains ([29, Theorem 1.1], [26, Theorem 1.1]): when $1 < p < \infty$, one can show that

$$ \min_{c \in \mathbb{R}} \|u - c\|_{L^p(\Omega)} \le C_{\text{EFNT},p} \delta(\Omega) \|\nabla u\|_{L^p(\Omega)}, \quad \forall u \in W^{1,p}(\Omega), \qquad (23) $$

where $C_{\text{EFNT},p}$ is the best possible constant that only depends on $p$ and equals

$$ C_{\text{EFNT},p} := \frac{p \sin(\pi/p)}{2\pi \sqrt{p-1}}. $$

Note that the last inequalities from (18) imply, again when $1 < p < \infty$, the Poincaré inequalities

$$ \|u - u_{\Omega}\|_{L^p(\Omega)} \le 2^{1-\frac{2}{p}} C_{\text{EFNT},p} \delta(\Omega) \|\nabla u\|_{L^p(\Omega)}, \quad \forall u \in W^{1,p}(\Omega). \qquad (24) $$

When $p = 1$, then the optimal Poincaré constant also bounds the Poincaré-Friedrichs constant:

$$ \min_{c \in \mathbb{R}} \|u - c\|_{L^1(\Omega)} \le \frac{\delta(\Omega)}{2} \|\nabla u\|_{L^1(\Omega)}, \quad \forall u \in W^{1,1}(\Omega). \qquad (25) $$

When $p = \infty$, since bounded convex domains are Lipschitz domains, Rademacher's theorem leads to

$$ \min_{c \in \mathbb{R}} \|u - c\|_{L^{\infty}(\Omega)} \le \delta(\Omega) \|\nabla u\|_{L^{\infty}(\Omega)}, \quad \forall u \in W^{1,\infty}(\Omega). \qquad (26) $$

**Remark 2.1.** Any estimate for the Poincaré-Friedrichs constant implies an estimate for the Poincaré constant, via (18). Let us compare $C_{\text{CW},p}$ for the Poincaré inequality with $C_{\text{EFNT},p}$ for the Poincaré-Friedrichs inequality. In the case $2 \le p$,

$$ 2^{1-\frac{2}{p}} C_{\text{EFNT},p} = \frac{2^{1-\frac{2}{p}} \sin(\pi/p)}{2 \pi/p} \frac{1}{\sqrt{p-1}} \le 4^{-\frac{1}{p}} \le C_{\text{CW},p}. $$

In the case $p \le 2$,

$$ 2^{\frac{2}{p}-1} C_{\text{EFNT},p} = \frac{2^{\frac{2}{p}-1} \sin(\pi/p)}{2 \pi/p} \frac{1}{\sqrt{p-1}} \le \frac{2^{\frac{2}{p}-1}}{2} = 2^{\frac{2}{p}-2} = 4^{\frac{1}{p}-1} \le C_{\text{CW},p}. $$

It follows that (24) is generally a tighter estimate than (21) for $1 < p < \infty$.

**Remark 2.2.** The above Poincaré and Poincaré-Friedrichs constants are optimal for the class of bounded convex domains, but individual bounded convex domains may allow for better constants. We refer to [53, 18, 55] for discussions; for example, triangles allow the reduction of the constant by 20%.

# 3 Basic notions of triangulations

We gather basic notions and definitions concerning simplicial meshes.

A $k$-dimensional simplex $T$ is the convex hull of $k+1$ affinely independent points $v_0, v_1, \dots, v_k \in \mathbb{R}^n$. We call these points the *vertices* of the simplex $T$. The strictly positive convex combinations of the vertices of the simplex constitute the *interior* of the simplex, and its remaining points constitute the *boundary* of the simplex. If $S$ is a simplex whose vertices are also vertices of another simplex $T$, in which case $S \subseteq T$, then we call $S$ a *subsimplex* of $T$ and call $T$ a *supersimplex* of $S$.

A finite family of simplices $\mathcal{T}$ is a *simplicial complex* or *triangulation* if it satisfies the following conditions: (i) $\mathcal{T}$ contains all the subsimplices of its members (ii) any non-empty intersection of two members of $\mathcal{T}$ is a common subsimplex of each. We say that a simplicial complex $\mathcal{T}$ has *dimension $n$* or is $n$-dimensional if each of its simplices is a subset of an $n$-dimensional member of that triangulation.<fnref n="3" />

<footnote n="3">³Simplicial complexes that we call $n$-dimensional are called purely $n$-dimensional in the literature on polytopes (cf. [71]) and simply “simplicial meshes” in the finite element literature.</footnote>
<figure><img src="image_3.png" /><figcaption>Figure 1: From left to right: local patches around a vertex, and edge, and a triangle. The local patch of any full-dimensional simplex only consists of that simplex itself (and its subsimplices).</figcaption></figure>

We also write $|\mathcal{T}|$ for the underlying set of the simplicial complex $\mathcal{T}$, which is the union $|\mathcal{T}| = \bigcup \mathcal{T}$ of all simplices in $\mathcal{T}$. One calls any set triangulable if it is the underlying set of some triangulation.

Given any simplex $T$, we write $S^{\downarrow}(T)$ for the simplicial complex that contains all subsimplices of $T$, and $S_k^{\downarrow}(T) \subseteq S^{\downarrow}(T)$ denotes the set of $k$-dimensional subsimplices of $T$. We write $\mathcal{V}(T) := S_0^{\downarrow}(T)$ for the set of vertices of $T$. Whenever $\mathcal{T}$ is a simplicial complex, the set of $k$-dimensional simplices in $\mathcal{T}$ is denoted as $S_k^{\downarrow}(\mathcal{T})$. Similarly, the notations $\mathcal{V}(\mathcal{T}) := S_0^{\downarrow}(\mathcal{T})$ and $\mathcal{F}(\mathcal{T}) := S_{n-1}^{\downarrow}(\mathcal{T})$ refer to the vertices and the faces (that is, members with codimension one) of this triangulation.<fnref n="4" /> In practice, we may identify points and singleton simplices.

When $\mathcal{T}$ is a triangulation and $T \in \mathcal{T}$, then $\mathrm{st}_{\mathcal{T}}(T)$ denotes the local patch or local star of $T$, which is the simplicial subcomplex of $\mathcal{T}$ that contains all supersimplices of $T$ and their subsimplices. We write $\partial\mathrm{st}_{\mathcal{T}}(T)$ for the subset of the local patch whose members do not contain $T$ itself. Formally,

$$ \mathrm{st}_{\mathcal{T}}(T) := \bigcup_{\substack{T' \in S_n^{\downarrow}(\mathcal{T}) \\ T \subseteq T'}} S^{\downarrow}(T'), \quad \partial\mathrm{st}_{\mathcal{T}}(T) := \bigcup_{\substack{T' \in \mathrm{st}_{\mathcal{T}}(T) \\ T \not\subseteq T'}} S^{\downarrow}(T'). $$

We also write $U_T := |\mathrm{st}_{\mathcal{T}}(T)|$ for the closed underlying set of the local patch. A crucial structural observation is the following.

**Lemma 3.1.** Let $\mathcal{T}$ be an $n$-dimensional simplicial complex and let $S, S' \in \mathcal{T}$. Then either $\mathrm{st}_{\mathcal{T}}(S)$ and $\mathrm{st}_{\mathcal{T}}(S')$ only intersect in simplices of dimension at most $n-1$ or there exists $S'' \in \mathcal{T}$ such that

$$ \mathrm{st}_{\mathcal{T}}(S) \cap \mathrm{st}_{\mathcal{T}}(S') = \mathrm{st}_{\mathcal{T}}(S''), \quad \mathcal{V}(S) \cup \mathcal{V}(S') = \mathcal{V}(S''). $$

*Proof.* Let $T \in \mathcal{T}$ be $n$-dimensional. We have $T \in \mathrm{st}_{\mathcal{T}}(S)$ if and only if all vertices of $S$ are vertices of $T$. Similarly, $T \in \mathrm{st}_{\mathcal{T}}(S')$ if and only if all vertices of $S'$ are vertices of $T$. Consequently, $T \in \mathrm{st}_{\mathcal{T}}(S) \cap \mathrm{st}_{\mathcal{T}}(S')$ if and only if $T \in \mathrm{st}_{\mathcal{T}}(S'')$, where $S'' \in \mathcal{T}$ is the convex closure of $S$ and $S'$. $\square$

We introduce a specific notion of connectivity when we are given an $n$-dimensional simplicial complex $\mathcal{T}$. We call two $n$-dimensional simplices $S, S' \in \mathcal{T}$ face-neighboring if $S \cap S'$ is a common face of both of them. We call $n$-simplices $S, S' \in \mathcal{T}$ face-connected in $\mathcal{T}$ if there exists a sequence $S = S_0, S_1, \dots, S_m = S' \in \mathcal{T}$ such that $S_i$ and $S_{i-1}$ are face-neighboring for all $1 \le i \le m$. Such a sequence is called a *face path* from $S$ to $S'$ in $\mathcal{T}$. Clearly, face-connected in $\mathcal{T}$ is an equivalence relation among simplices. A *face-connected component* of $\mathcal{T}$ is an equivalence class under this equivalence relation, and we call $\mathcal{T}$ *face-connected* if it has only one face-connected component.

## 3.1 Shape measures and related quantities

We introduce several quantities that measure the regularity of a triangulation. These have in common that they can be computed from purely local information.

We write $\delta(T)$ and $\mathrm{vol}(T)$ for the diameter and $n$-dimensional volume of any $n$-simplex $T$. Moreover, $h(T)$ refers to the smallest height of any vertex of the simplex $T$, where the height of a vertex is defined

<footnote n="4">⁴Our use of the term *face* as is common in classical geometry and the finite element literature [13] and is synonymous with *facet* as used in the literature on polyhedral combinatorics [62]. Notably, this terminology differs from the uses *face* and *facet* in the theory of polyhedra [71].</footnote>
as the distance to the affine span of its opposing face. For the purpose of the usual scaling arguments, the n-dimensional reference simplex $\Delta^n \subseteq \mathbb{R}^n$ is the convex closure of the origin and the n canonical unit vectors.

Whenever $T$ is any $n$-dimensional simplex $T$, we define the aspect shape measure $\kappa_A(T)$, and the algebraic shape measure $\kappa_M(T)$ by

$$ \kappa_A(T) := \frac{\delta(T)}{h(T)}, \qquad \kappa_M(T) := \sup_{\varphi: \Delta^n \to T} \|\mathbf{J}\varphi\|_2 \|\mathbf{J}\varphi^{-1}\|_2, \qquad (27) $$

where the last supremum is taken over all affine transformation from the reference $n$-simplex onto the $n$-simplex $T$. When $\mathcal{T}$ is an $n$-dimensional simplicial complex, we naturally define

$$ \kappa_A(\mathcal{T}) := \sup_{T \in \mathcal{S}_n^+(\mathcal{T})} \kappa_A(T), \qquad \kappa_M(\mathcal{T}) := \sup_{T \in \mathcal{S}_n^+(\mathcal{T})} \kappa_M(T). \qquad (28) $$

We call these the aspect and algebraic shape measure, respectively, of the triangulation.

**Remark 3.2.** The ratio $\kappa_A(T)$ measures the “shape quality” of an $n$-dimensional simplex $T$ and is an instance of a so-called shape measure. For example, the reference triangle has aspect shape measure 2 and the reference tetrahedron has aspect shape measure $\sqrt{6}$. Numerous alternative shape measures have been used throughout the literature of numerical analysis and computational geometry to quantify the quality of simplices (see [11, p.61, Definition 5.1], [13, p.97, Definition (4.2.16)], [24, Definition 11.2]).

We gather a few relationships between geometric and algebraic entities and compare the different shape measures of a single simplex.

**Lemma 3.3.** Let $T$ be an $n$-simplex and let $\varphi: \Delta^n \to T$ be an affine diffeomorphism from the reference $n$-simplex. Then

$$ \|\mathbf{J}\varphi\|_2 \le C_{1,n} \cdot \delta(T), \qquad \|\mathbf{J}\varphi^{-1}\|_2 \le C_{2,n} \cdot \kappa_A(T)\delta(T)^{-1}, \\ \frac{1}{\sqrt{2n}} \kappa_A(T) \le \kappa_M(T) \le n\kappa_A(T). $$

Here, $C_{1,n}, C_{2,n} \le \sqrt{n}$.

*Proof.* Let $\varphi: \Delta^n \to T$ be an affine transformation. We abbreviate $J := \mathbf{J}\varphi$ for its Jacobian. We begin with observing that the largest $\ell^2$-norm of any column of $J$, which we denote here by $c_{\max}(J)$, equals the maximum of the quotient $\|Jx\|_{\ell^2}/\|x\|_{\ell^1}$ over all non-zero $x \in \mathbb{R}^n$. The diameter of $T$ is the length of its longest edge. The first inequality is implied by the following comparisons:

$$ \frac{\delta(T)}{\sqrt{2}} \le \|J\|_2 \le \sqrt{n} \cdot c_{\max}(J) \le \sqrt{n} \cdot \delta(T). $$

The columns of the matrix $J^{-1}$ are the gradients of the barycentric coordinates of the vertices of $T$, except for the vertex $\varphi(0) \in T$. It immediately follows that

$$ \|J^{-1}\|_2 \le \sqrt{n} \cdot c_{\max}(J^{-1}) \le \sqrt{n} \cdot h(T)^{-1}. $$

The smallest height in the reference simplex $\Delta^n$ is $h_\Delta = 1/\sqrt{n}$, whence $h(T) \ge \|J^{-1}\|_2^{-1}/\sqrt{n}$. This yields the second inequality. Lastly, the last comparison is now obvious. $\square$

We will also need the maximal ratio of volumes between face-neighboring $n$-simplices, written $C_\rho(\mathcal{T})$, and the ratio of the diameters of any intersecting simplices, written $C_\theta(\mathcal{T})$. Formally,

$$ C_{\rho}(\mathcal{T}) := \sup_{\substack{T, T' \in \mathcal{S}_n^+(\mathcal{T}) \\ T \cap T' \neq \emptyset}} \frac{\mathrm{vol}(T)}{\mathrm{vol}(T')}, \qquad (29) $$

$$ C_{\theta}(\mathcal{T}) := \sup_{\substack{T, T' \in \mathcal{S}_n^+(\mathcal{T}) \\ T \cap T' \neq \emptyset}} \frac{\delta(T)}{\delta(T')}. \qquad (30) $$
Finally, whenever $T, T'$ are two $n$-simplices that share a common face $F$ of codimension 1, we let $\Xi_{T,T'} : T \to T'$ denote the affine diffeomorphism that preserves $F$. We then define

$$C_{\xi}(T) := \sup_{\substack{T, T' \in S_n^{\perp}(T) \\ T \cap T' \in S_{n-1}^{\perp}(T)}} \|\mathbf{J}\Xi_{T, T'}\|_2 \quad (31)$$

to be the maximum of the operator norm of the Jacobian of any such diffeomorphism. This indicator quantifies how much reflection across the shared face distorts the geometry.

**Lemma 3.4.** Let $T_1$ and $T_2$ be two $n$-simplices that share a common face $F$. Then

$$\delta(T_1) \le \kappa_A(T_1)\delta(F) \le \kappa_A(T_1)\delta(T_2), \quad \frac{\mathrm{vol}(T_1)}{\mathrm{vol}(T_2)} \le \frac{\delta(T_1)}{\delta(T_2)}\kappa_A(T_2) \le \kappa_A(T_1)\kappa_A(T_2).$$

If $\Xi: T_1 \to T_2$ is the affine diffeomorphism that is the identity over $F$, then at least $n-2$ of its Jacobian's singular values equal 1, and we have

$$\begin{align*}
\|\mathbf{J}\Xi\|_2 &\le \frac{1}{2} \sqrt{\left(\frac{\delta(T_2)}{\delta(T_1)}\kappa_A(T_1) + 1\right)^2 + \kappa_A(T_1)^2 + \frac{1}{2}\sqrt{\left(\frac{\delta(T_2)}{\delta(T_1)}\kappa_A(T_1) - 1\right)^2 + \kappa_A(T_1)^2}}, \\
\|\mathbf{J}\Xi^{-1}\|_2 &\le \frac{1}{2} \sqrt{\left(\frac{\delta(T_1)}{\delta(T_2)}\kappa_A(T_2) + 1\right)^2 + \kappa_A(T_2)^2 + \frac{1}{2}\sqrt{\left(\frac{\delta(T_1)}{\delta(T_2)}\kappa_A(T_2) - 1\right)^2 + \kappa_A(T_2)^2}}, \\
\det(\mathbf{J}\Xi) &= \frac{\mathrm{vol}(T_2)}{\mathrm{vol}(T_1)} \le \frac{\delta(T_2)}{\delta(T_1)}\kappa_A(T_1).
\end{align*}$$

*Proof.* The diameter of $F$ is at least as large as the height $h_S$ of some other vertex of $F$ in $T_1$. To see this, recall that the height of a vertex is minimum distance to the affine hull of the opposing subsimplex. Naturally, the height of the vertex in $T_1$ will be smaller than the height in $F$. Now,

$$\delta(T_1)\kappa_A(T_1)^{-1} = h(T_1) \le h_S \le \delta(F).$$

The first estimate follows. As for the second estimate, let $h_1, h_2 > 0$ be the heights of $F$ in the simplices $T_1$ and $T_2$, respectively. By the volume formula for simplices, $\mathrm{vol}(T_1) = h_1 \mathrm{vol}(F)/n$ and $\mathrm{vol}(T_2) = h_2 \mathrm{vol}(F)/n$, and thus $\mathrm{vol}(T_1)/\mathrm{vol}(T_2) = h_1/h_2$. Thus follows the second estimate:

$$\frac{\mathrm{vol}(T_1)}{\mathrm{vol}(T_2)} = \frac{h_1}{h_2} \le \frac{\delta(T_1)}{h_2} \le \kappa_A(T_1) \frac{\delta(F)}{h_2} \le \kappa_A(T_1) \kappa_A(T_2), \quad \frac{\delta(T_1)}{h_2} = \frac{\delta(T_1)}{\delta(T_2)} \kappa_A(T_2).$$

Lastly, let $\Xi: T_1 \to T_2$ be as stated. To estimate the Lipschitz constant of $\Xi$, we study the singular values of its Jacobian. Without loss of generality, $F$ lies in the span of the first $n-1$ coordinates and contains the origin. Write $z_1 \in T_1$ and $z_2 \in T_2$ for the two vertices not contained in $F$. There exists a unit height vector $\hat{h}_0$ on the $n$-th coordinate direction such that there exist $b_1, b_2 \in \mathbb{R}^n$ in the affine hull of $F$ satisfying $z_1 = b_1 + h_1\hat{h}_0$ and $z_2 = b_2 - h_2\hat{h}_0$. The mapping $\Xi$ is linear, being the identity over $F$ and mapping $z_1$ to $z_2$. Hence,

$$\Xi(\hat{h}_0) = h_1^{-1}(b_2 - b_1) - h_2 h_1^{-1} \hat{h}_0.$$

We see that $\Xi$ equals the identity over the orthogonal complement of the span of $\hat{h}_0$ and $b_2 - b_1$. The only singular values of its Jacobian are the two singular values $\sigma_- \le 1 \le \sigma_+$ of the matrix

$$\begin{pmatrix} a & 0 \\ c & 1 \end{pmatrix}, \quad a = -h_2/h_1, \quad c = \|b_2 - b_1\|/h_1.$$

These are

$$\begin{align*}
\sigma_{\pm} &= \frac{1}{\sqrt{2}} \sqrt{1 + a^2 + c^2 \pm \sqrt{(1 + a^2 + c^2)^2 - 4a^2}} \\
&= \frac{1}{\sqrt{2}} \sqrt{1 + a^2 + c^2 \pm \sqrt{((1 + a)^2 + c^2)((1 - a)^2 + c^2)}} \\
&= \frac{1}{2} \left( \sqrt{(1 + a)^2 + c^2} \pm \sqrt{(1 - a)^2 + c^2} \right).
\end{align*}$$
We notice that this provides the upper bound

$$ \sigma_+ \le \frac{1}{2} \sqrt{ \left( \frac{\delta(T_2)}{\delta(T_1)} \kappa_A(T_1) + 1 \right)^2 + \kappa_A(T_1)^2 } + \frac{1}{2} \sqrt{ \left( \frac{\delta(T_2)}{\delta(T_1)} \kappa_A(T_1) - 1 \right)^2 + \kappa_A(T_1)^2 } $$

The desired estimates are shown.

<figure><img src="image_4.png" /></figure>

**Remark 3.5.** While we will utilize $C_\theta(T)$ at numerous places throughout the manuscript, $C_\rho(T)$ and $C_\xi(T)$ will only be used throughout the following Section 4. Lemma 3.4 expresses that $C_\rho(T)$ of (29) is controlled by the shape measure. In a face-connected triangulation where we have an upper bound for the number of simplices sharing a vertex, this Lemma also indicates, at least in principle, control of $C_\theta(T)$ of (30).

# 4 Poincaré–Friedrichs inequalities over triangulated domains

In this section, we develop stepwise computable estimates for Poincaré–Friedrichs constants of triangulated domains. The following very classical procedure serves as our inspiration: given a gradient vector field, we can reconstruct a gradient potential by fixing a starting point and integrating the gradient field along lines emanating from that point; the potential is unique up to an additive constant. We perform a discrete analogue of this procedure over triangulated domains: having fixed a starting simplex, we first define the scalar potential on this simplex. Step by step, we traverse the triangulation along face-neighboring simplices, each time extending the scalar potential to the next simplex and controlling its Lebesgue norm. As we construct the scalar potential over increasingly larger subdomains, we always pick a gradient potential on the next simplex and fix the integration constant using the value already known along the connecting face, thereby ensuring the correct Sobolev continuity. This basic idea has appeared in various forms before, for instance recently in [12, 25, 19, 68].

We begin with an auxiliary result with independent relevance, where we estimate the Poincaré–Friedrichs inequality when homogeneous boundary conditions hold along at least one face of the boundary.

**Lemma 4.1.** Let $T$ be an $n$-simplex with a face $F$ and $p \in [1, \infty]$. If $u \in W^{1,p}(T)$ with $\text{tr}_F u = 0$, then

$$ \|u\|_{L^p(T)} \le C_{\text{PF},T,F,p} \|\nabla u\|_{L^p(T)}, $$

where $C_{\text{PF},T,F,p} = p^{-\frac{1}{p}}\delta(T)$ for $p < \infty$ and $C_{\text{PF},T,F,\infty} = \delta(T)$.

*Proof.* Since the inequality follows from Rademacher's theorem in the limit case $p = \infty$, we assume $1 \le p < \infty$. Let $u \in C^\infty(T)$ have support disjoint from $F$. We tacitly extend this by zero to a function $u \in L^\infty(\mathbb{R}^n)$. Without loss of generality, the segment from the midpoint of $F$ to the opposing vertex lies on the first coordinate axis, and the minimal first coordinate among all the points of $F$ equals 0. We write $g$ for the trivial extension of $\nabla u$ over the entire $\mathbb{R}^n$. Using the fundamental theorem of calculus and Hölder's inequality,

$$ \begin{aligned} \int_T |u(x)|^p dx \, dx &\le \int_{\mathbb{R}^{n-1}} \int_0^{\delta(T)} |u(x_1, \bar{x})|^p dx_1 \, d\bar{x} \\ &\le \int_{\mathbb{R}^{n-1}} \int_0^{\delta(T)} \left| \int_0^{x_1} |g(y, \bar{x})| dy \right|^p dx_1 \, d\bar{x} \\ &\le \int_0^{\delta(T)} x_1^{p-1} \int_{\mathbb{R}^{n-1}} \int_0^{x_1} |g(y, \bar{x})|^p dy \, d\bar{x} \, dx_1 \\ &\le \int_0^{\delta(T)} x_1^{p-1} dx_1 \cdot \int_T |g(y, \bar{x})|^p dy \, d\bar{x} \le \frac{\delta(T)^p}{p} \int_T |\nabla u(x)|^p dx. \end{aligned} $$

If $u \in W^{1,p}(T)$ has vanishing trace along $F$ but is not necessarily smooth, then we conclude $\|u\|_{L^p(T)} \le \delta(T)p^{-\frac{1}{p}} \|\nabla u\|_{L^p(T)}$ from approximation via members of $C^\infty(T)$ whose support is disjoint from $F$. We very briefly verify that density argument: There exists an affine diffeomorphism $\varphi: \Delta^n \to T$ from the reference simplex onto $T$ that maps the convex closure of the $n$ unit vectors onto the face $F$. We let
$\hat{u} := u \circ \varphi$. Let $\hat{U}$ be the unit ball of the $l^1$ metric, which contains $\Delta^n$. We let $\tilde{u}$ be the extension of $\hat{u}$ onto $\hat{U}$ by reflection across the coordinate axes. Then $\tilde{u} \in W_0^{1,p}(\hat{U})$, and $\tilde{u}$ is the limit of a sequence $u_m \in C_c^\infty(\hat{U})$. Now $u_m \circ \varphi^{-1} \in C^\infty(T)$ approximates $u$ within the Banach space $W^{1,p}(T)$ and has the desired support property. $\square$

**Remark 4.2.** We can improve Lemma 4.1 in the special case $p = 2$. The variational formulation of the Poincaré constant over a convex domain reveals that $C_{PF,T,F,p}$ lies between the Poincaré constant without boundary conditions and with full boundary conditions. In particular,

$$C_{PF,T,F,2} \leq \frac{\delta(T)}{\pi}$$

is an improved Poincaré inequality.

The next auxiliary result establishes Poincaré-Friedrichs constants over face patches within simplicial triangulations. We emphasize that face patches are not necessarily convex, but we can still extend the results on convex domains from Section 2.2. The following result is reasonably sharp when the two simplices have similar volumes and diameters.

**Lemma 4.3.** Let $\mathcal{T}$ be a triangulation. Let $T_1, T_2 \in \mathcal{T}$ be two $n$-simplices whose intersection is a common face $F := T_1 \cap T_2$. Write $U_F := T_1 \cup T_2$. If $p \in [1, \infty]$ and $u \in W^{1,p}(\Omega)$, then

$$\min_{c \in \mathbb{R}} \|u - c\|_{L^p(U_F)} \leq C_{PF, T_1 \cup T_2, p} \|\nabla u\|_{L^p(U_F)}.$$

Here,

$$C_{PF,T_1 \cup T_2, p} \leq 2C_{1,n}C_{EFNT,p}C_{\rho}(\mathcal{T})^{\frac{1}{p}} \max(\delta(T_1), \delta(T_2)).$$

If $U_F$ is convex, then $C_{PF,T_1 \cup T_2, p} \leq C_{EFNT,p} \delta(U_F)$.

*Proof.* The convex case follows immediately from (23). Otherwise, without loss of generality, $F$ has the vertices $v_0, \dots, v_{n-1}$, and $z_1 \in T_1$ and $z_2 \in T_2$ are the remaining vertices of the two triangles. Let $\Delta_1 = \Delta^n$ be the reference $n$-simplex and let $\Delta_2$ be obtained from it by flipping the $n$-th coordinate. We let $\varphi_1 : \Delta_1 \to T_1$ and $\varphi_2 : \Delta_2 \to T_2$ be affine transformations that map the origin to $v_0$, that map each unit vector $e_i$ to $v_i$ for $i = 1, \dots, n-1$, and that satisfy $\varphi_1(e_n) = z_1$ and $\varphi_2(-e_n) = z_2$. Write $\hat{U} := \Delta_1 \cup \Delta_2$. We have a bi-Lipschitz mapping $\varphi : \hat{U} \to U_F$.

Suppose that $u \in W^{1,p}(U_F)$. Then $\hat{u} := u \circ \varphi \in W^{1,p}(\hat{U})$. We observe

$$\|\nabla \hat{u}\|_{L^p(\hat{U})} \leq \max \left( |\det(\mathbf{J}\varphi_1)|^{-\frac{1}{p}} \|\mathbf{J}\varphi_1\|_2, |\det(\mathbf{J}\varphi_2)|^{-\frac{1}{p}} \|\mathbf{J}\varphi_2\|_2 \right) \|\nabla u\|_{L^p(U_F)}.$$

Notice that $\hat{U}$ has diameter 2 and is (crucially) convex. Thus, due to Poincaré-Friedrichs inequality (23), there exists $\hat{w} \in W^{1,p}(\hat{U})$ such that $\nabla \hat{w} = \nabla \hat{u}$ and

$$\|\hat{w}\|_{L^p(\hat{U})} \leq 2C_{EFNT,p} \|\nabla \hat{u}\|_{L^p(\hat{U})}.$$

Next, setting $w := \hat{w} \circ \varphi^{-1}$, we find $\nabla w = \nabla u$ and

$$\|w\|_{L^p(U_F)} \leq \max \left( |\det(\mathbf{J}\varphi_1)|, |\det(\mathbf{J}\varphi_2)| \right)^{\frac{1}{p}} \|\hat{w}\|_{L^p(\hat{U})}.$$

For both $i = 1, 2$, we now recall the well-known equation $|\det(\mathbf{J}\varphi_i)| = n! \text{vol}(T_i)$, and the estimate $\|\mathbf{J}\varphi_i\|_2 \leq C_{1,n} \delta(T_i)$, which is given in Lemma 3.3. We obtain the desired result. $\square$

The main result of this section constructs a potential and gives an upper bound for the Poincaré-Friedrichs constant. It follows the same underlying principle as the “discrete mean Poincaré inequality” of [27, Lemma 3.7]. This procedure serves as the blueprint for constructing potentials of the curl and divergence operators in later sections.

Two different variations of the underlying idea are analyzed, yielding slightly different estimates. In the sequential sweep over the simplices in the triangulation $\mathcal{T}$, we can either employ Poincaré-Friedrichs inequalities over the individual simplices while relying on gluing with the previously visited face neighbor via Lemma 4.1, or we consider the face patch (star) formed by the current and the previous simplex and employ Lemma 4.3. Both estimates of Poincaré-Friedrichs constants capture the correct asymptotic behavior as $p$ grows to infinity. We first prepare a recursion result in the forthcoming Theorem 4.4 that we subsequently convert in our final estimate in Theorem 4.5.
Figure 2: Face-connected triangulation of a domain. The arrows depict a spanning tree in the face-connection graph.

<figure><img src="image_5.png" /></figure>

**Theorem 4.4.** Let $\mathcal{T}$ be a face-connected $n$-dimensional finite triangulation and let the domain $\Omega$ be the interior of the underlying set of $\mathcal{T}$. Let $1 \le p, q \le \infty$ with $1 = 1/p + 1/q$. Then for any $u \in W^{1,p}(\Omega)$ there exists $w \in W^{1,p}(\Omega)$ with $\nabla w = \nabla u$ and satisfying the following estimates:

1. There exists an $n$-simplex $T_0 \in \mathcal{T}$ with

$$ \|w\|_{L^p(T_0)} \le C_{\mathrm{PF},T_0,p} \|\nabla u\|_{L^p(T_0)}. \quad (32) $$

2. For any $n$-simplex $T_M \in \mathcal{T}$, there exists a face path $T_0, T_1, \dots, T_M$ such that for all $1 \le m \le M$, we have the two recursive estimates:

$$ \|w\|_{L^p(T_m)} \le A_m \|w\|_{L^p(T_{m-1})} + B'_m \|\nabla u\|_{L^p(T_m)} + B''_m \|\nabla u\|_{L^p(T_{m-1})}, \quad (33) $$

$$ \|w\|_{L^p(T_m)} \le A_m \|w\|_{L^p(T_{m-1})} + B_m^* \|\nabla u\|_{L^p(U_{F_m})}, \quad (34) $$

with the constants

$$ \begin{aligned} A_m \le C_{\rho}(\mathcal{T})^{\frac{1}{p}}, \quad B'_m \le C_{\mathrm{PF},T_m,F_m,p}, \quad B''_m \le C_{\mathrm{PF},T_m,F_m,p} C_{\rho}(\mathcal{T})^{\frac{1}{p}} C_{\xi}(\mathcal{T}), \\ B_m^* \le \left(1 + C_{\rho}(\mathcal{T})^{\frac{q}{p}}\right)^{\frac{1}{q}} C_{\mathrm{PF},U_{F_m},p}. \end{aligned} $$

Here, $F_m := T_m \cap T_{m-1}$ and $U_m := T_m \cup T_{m-1}$ for any $1 \le m \le M$.

*Proof.* Let $u \in W^{1,p}(\Omega)$. We start with the Poincaré-Friedrichs inequality on the first simplex $T_0$. Via (14), there exists $w_0 \in W^{1,p}(T_0)$ satisfying $\nabla w_0 = \nabla u$ over $T_0$ together with

$$ \|w_0\|_{L^p(T_0)} \le C_{\mathrm{PF},T_0,p} \|\nabla u\|_{L^p(T_0)}. $$

In particular, $c_0 := (w_0 - u)|_{T_0}$ is a constant function. We then define

$$ w := u + c_0 $$

as our potential. Clearly, $w \in W^{1,p}(\Omega)$ with $\nabla w = \nabla u$. By construction $w|_{T_0} = w_0$ and (32) holds. In the rest of the proof, we verify that $w$ satisfies the desired recursive estimates (33) and (34). Suppose that $T_0, T_1, \dots, T_M$ is a face path in $\mathcal{T}$ and that $1 \le m \le M$. Recall that we write $F_m := T_m \cap T_{m-1}$, which is a face of dimension $n-1$ shared by the $n$-simplices $T_m$ and $T_{m-1}$, and that we write $U_{F_m} := T_m \cup T_{m-1}$.

We study two constructions, beginning as follows. We define $w'_m := w|_{T_{m-1}} \circ \Xi \in W^{1,p}(T_m)$, where $\Xi : T_m \to T_{m-1}$ is the unique affine diffeomorphism that leaves $F_m$ invariant. By construction, $w'_m \in W^{1,p}(T_m)$ with

$$ \mathrm{tr}_{F_m} w'_m = \mathrm{tr}_{F_m} w|_{T_{m-1}}. $$

We now define $w''_m \in W^{1,p}(T_m)$ via

$$ w''_m := w|_{T_m} - w'_m = u|_{T_m} - u|_{T_{m-1}} \circ \Xi. \quad (35) $$
We crucially note that $w_m''$ is trace-free along $F_m$ since

$$ \mathrm{tr}_{F_m} w_m'' = \mathrm{tr}_{F_m} (w|_{T_m} - w_m') = \mathrm{tr}_{F_m} w|_{T_m} - \mathrm{tr}_{F_m} w|_{T_{m-1}} = \mathrm{tr}_{F_m} u|_{T_m} - \mathrm{tr}_{F_m} u|_{T_{m-1}} = 0. $$

An application of Lemma 4.1 to the first expression in (35) gives

$$ \begin{aligned} \|w_m''\|_{L^p(T_m)} &\le C_{\mathrm{PF},T_m,F_m,p} (\|\nabla w\|_{L^p(T_m)} + \|\nabla w_m'\|_{L^p(T_m)}) \\ &= C_{\mathrm{PF},T_m,F_m,p} (\|\nabla u\|_{L^p(T_m)} + \|\nabla w_m'\|_{L^p(T_m)}). \end{aligned} $$

Using Lemma 3.4 as well as Definitions (29) and (31), we find

$$ \begin{aligned} \|\nabla w_m'\|_{L^p(T_m)} &\le |\det(\mathbf{J}\Xi)|^{-\frac{1}{p}} \|\mathbf{J}\Xi\|_2 \|\nabla w\|_{L^p(T_{m-1})} \\ &\le \left( \frac{\mathrm{vol}(T_m)}{\mathrm{vol}(T_{m-1})} \right)^{\frac{1}{p}} C_{\xi}(T) \|\nabla w\|_{L^p(T_{m-1})} = C_{\rho}(T)^{\frac{1}{p}} C_{\xi}(T) \|\nabla u\|_{L^p(T_{m-1})}. \end{aligned} $$

Since $w|_{T_m} = w_m'' + w_m'$, we finally find

$$ \begin{aligned} \|w\|_{L^p(T_m)} &\le \|w_m'\|_{L^p(T_m)} + \|w_m''\|_{L^p(T_m)} \\ &\le C_{\rho}(T)^{\frac{1}{p}} \|w\|_{L^p(T_{m-1})} + C_{\mathrm{PF},T_m,F_m,p} \left( \|\nabla u\|_{L^p(T_m)} + C_{\rho}(T)^{\frac{1}{p}} C_{\xi}(T) \|\nabla u\|_{L^p(T_{m-1})} \right). \end{aligned} $$

Therefrom, the first recursive estimate follows.

Now we discuss the second recursive estimate. Suppose again that $1 \le m \le M$. We use the Poincaré-Friedrichs inequality over $U_{F_m}$, as given in Lemma 4.3, to find $w_{F_m} \in W^{1,p}(U_{F_m})$ such that $\nabla w_{F_m} = \nabla u$ over $U_{F_m}$ and

$$ \|w_{F_m}\|_{L^p(U_{F_m})} \le C_{\mathrm{PF},U_{F_m},p} \|\nabla w_{F_m}\|_{L^p(U_{F_m})}. \quad (36) $$

We can define the constant

$$ c_m := w|_{U_{F_m}} - w_{F_m}. $$

Now we observe that

$$ \begin{aligned} \|w\|_{L^p(T_m)} &\le \|w_{F_m}\|_{L^p(T_m)} + \|c_m\|_{L^p(T_m)}, \\ \|c_m\|_{L^p(T_m)} &= \frac{\mathrm{vol}(T_m)^{\frac{1}{p}}}{\mathrm{vol}(T_{m-1})^{\frac{1}{p}}} \|c_m\|_{L^p(T_{m-1})}, \\ \|c_m\|_{L^p(T_{m-1})} &\le \|w\|_{L^p(T_{m-1})} + \|w_{F_m}\|_{L^p(T_{m-1})}. \end{aligned} $$

In combination,

$$ \|w\|_{L^p(T_m)} \le \|w_{F_m}\|_{L^p(T_m)} + \frac{\mathrm{vol}(T_m)^{\frac{1}{p}}}{\mathrm{vol}(T_{m-1})^{\frac{1}{p}}} \|w_{F_m}\|_{L^p(T_{m-1})} + \frac{\mathrm{vol}(T_m)^{\frac{1}{p}}}{\mathrm{vol}(T_{m-1})^{\frac{1}{p}}} \|w\|_{L^p(T_{m-1})}. $$

We sum the two integrals of $w_{F_m}$. When $1 < p < \infty$, recalling the complementary exponent $q = p/(p-1) \in (1, \infty)$, we use Hölder's inequality to verify

$$ \begin{aligned} \|w\|_{L^p(T_m)} &\le \left(1 + \frac{\mathrm{vol}(T_m)^{\frac{q}{p}}}{\mathrm{vol}(T_{m-1})^{\frac{q}{p}}}\right)^{\frac{1}{q}} \left(\|w_{F_m}\|_{L^p(T_m)}^p + \|w_{F_m}\|_{L^p(T_{m-1})}^p\right)^{\frac{1}{p}} + \frac{\mathrm{vol}(T_m)^{\frac{1}{p}}}{\mathrm{vol}(T_{m-1})^{\frac{1}{p}}} \|w\|_{L^p(T_{m-1})} \\ &= \left(1 + \frac{\mathrm{vol}(T_m)^{\frac{q}{p}}}{\mathrm{vol}(T_{m-1})^{\frac{q}{p}}}\right)^{\frac{1}{q}} \|w_{F_m}\|_{L^p(U_{F_m})} + \frac{\mathrm{vol}(T_m)^{\frac{1}{p}}}{\mathrm{vol}(T_{m-1})^{\frac{1}{p}}} \|w\|_{L^p(T_{m-1})}. \end{aligned} $$

Note that in the limit cases $p = 1$ and $p = \infty$ we get, respectively,

$$ \begin{aligned} \|w\|_{L^1(T_m)} &\le \max \left(1, \frac{\mathrm{vol}(T_m)}{\mathrm{vol}(T_{m-1})}\right) \|w_{F_m}\|_{L^1(U_{F_m})} + \frac{\mathrm{vol}(T_m)}{\mathrm{vol}(T_{m-1})} \|w\|_{L^1(T_{m-1})}, \\ \|w\|_{L^\infty(T_m)} &\le 2 \|w_{F_m}\|_{L^\infty(U_{F_m})} + \|w\|_{L^\infty(T_{m-1})}. \end{aligned} $$

The local inequality (36) now provides the second recursive estimate. The proof is complete. $\square$
The recursive construction of the gradient potential in the previous theorem, marching from simplex to simplex, can be associated with a concept of graph theory. Indeed, the face-neighbor relationship between adjacent n-simplices gives rise to an undirected graph that we call *face-connection graph*. Starting from an initial n-simplex $T_0$, any sequence of simplices corresponds to a path in that graph. In practice, we will pick a spanning tree for the undirected graph to describe the construction of the potentials.

We now use this formalism to describe an estimate for the Poincaré-Friedrichs constant of the gradient potential. The formulation of the result is somewhat technical, but the underlying idea is as follows. Unwrapping the recursion, we estimate the potential over a simplex in terms of the norm of the gradient vector field over numerous other simplices. Having estimated the potential on each simplex, a global estimate follows naturally and implies the desired Poincaré-Friedrichs inequality.

**Theorem 4.5.** Let $\mathcal{T}$ be a face-connected n-dimensional finite triangulation and let the domain $\Omega$ be the interior of the underlying set of $\mathcal{T}$. Let $1 \le p, q \le \infty$ with $1 = 1/p + 1/q$, and suppose further that $u, w \in W^{1,p}(\Omega)$ with $\nabla w = \nabla u$.

Suppose that we have an estimate of the form

$$ \|w\|_{L^p(T_0)} \le A_0 \|\nabla u\|_{L^p(T_0)}. $$

Suppose that the n-simplices are enumerated as $T_0, T_1, \dots, T_M$, and for each $1 \le m \le M$ we have $0 \le \varpi(m) \le m - 1$ such that

$$ \|w\|_{L^p(T_m)} \le A_m \|\nabla u\|_{L^p(T_m)} + A'_m \|\nabla u\|_{L^p(T_{\varpi(m)})} + B_m \|w\|_{L^p(T_{\varpi(m)})}, \quad 1 \le m \le M. $$

Then

$$ \|w\|_{L^p(\Omega)} \le \left( \sum_{m=0}^{M} \left( \sum_{\ell=0}^{L(m)} C_{m,\ell}^q \right)^{\frac{p}{q}} \right)^{\frac{1}{p}} \|\nabla u\|_{L^p(\Omega)}, $$

where $C_{m,\ell} \ge 0$ is as in the proof, with the obvious modifications if $p = 1$ or $p = \infty$.

*Proof.* Suppose that $1 \le m \le M$. Starting from index $m$, we repeatedly apply the predecessor function $\varpi$ until we come back to index zero. This produces a strictly ascending sequence of indices

$$ 0 = J_{m,0}, J_{m,1}, \dots, J_{m,L(m)} = m \qquad (37) $$

for some $L(m) \ge 1$ such that $J_{m,\ell-1} = \varpi(J_{m,\ell})$. Unwrapping the recursion for the norm over the m-th simplex, the total expression is now written in the form

$$ \begin{aligned} \|w\|_{L^p(T_m)} &\le \sum_{\ell=0}^{L(m)} (B_{J_{m,L(m)}} \cdots B_{J_{m,\ell+1}}) A_{J_{m,\ell}} \|\nabla u\|_{L^p(T_{J_{m,\ell}})} \\ &\quad + \sum_{\ell=0}^{L(m)-1} (B_{J_{m,L(m)}} \cdots B_{J_{m,\ell+2}}) A'_{J_{m,\ell+1}} \|\nabla u\|_{L^p(T_{J_{m,\ell}})} \\ &= A_m \|\nabla u\|_{L^p(T_{J_{m,L(m)}})} \\ &\quad + \sum_{\ell=0}^{L(m)-1} (B_{J_{m,L(m)}} \cdots B_{J_{m,\ell+2}}) (B_{J_{m,\ell+1}} A_{J_{m,\ell}} + A'_{J_{m,\ell+1}}) \|\nabla u\|_{L^p(T_{J_{m,\ell}})} \\ &=: \sum_{\ell=0}^{M} C_{m,\ell} \|\nabla u\|_{L^p(T_\ell)}. \end{aligned} $$

Here, $C_{m,\ell}$ is the coefficient of $\|\nabla u\|_{L^p(T_\ell)}$, possibly zero, as it appears in the unwrapped recursive
estimate of $||w||_{L^p(T_m)}$. The global Poincaré-Friedrichs inequality follows via Hölder's inequality:

$$
\begin{align*}
\|w\|_{L^p(\Omega)}^p &\le \sum_{m=0}^{M} \|w\|_{L^p(T_m)}^p \\
&\le \sum_{m=0}^{M} \left( \sum_{\ell=0}^{M} C_{m,\ell} \|\nabla u\|_{L^p(T_\ell)} \right)^p \\
&\le \sum_{m=0}^{M} \left( \sum_{\ell=0}^{M} C_{m,\ell}^q \right)^{\frac{p}{q}} \sum_{\ell'=0}^{M} \|\nabla u\|_{L^p(T_{\ell'})}^p \le \left( \sum_{m=0}^{M} \left( \sum_{\ell=0}^{M} C_{m,\ell}^q \right)^{\frac{p}{q}} \right) \|\nabla u\|_{L^p(\Omega)}^p,
\end{align*}
$$

where $q \in [1, \infty]$ satisfies $1 = 1/p + 1/q$ and with obvious modifications if $p = 1$ or $p = \infty$. $\square$

**Remark 4.6.** The computable Poincaré-Friedrichs constants obtained in Theorem 4.5 depend on only a few parameters of the given triangulation: the length of any traversal from the root simplex, the volume ratios of any pair of adjacent simplices along this path, and the Poincaré-Friedrichs constants on each simplex (Lemma 4.1) or on face patch (Lemma 4.3), then possibly including shape regularity parameters.

These computable Poincaré-Friedrichs constants increasingly overestimate the best one as the number of n-simplices in the triangulation $\mathcal{T}$ increases. Hence, we conceive their main application to be local patches (stars), in particular non-convex boundary stars. The latter occur inevitably at reentrant corners. Clearly, the same building principle applies whenever we have any non-overlapping partition of $\bar{\Omega}$ into convex local patches $\{U_m\}$ of n-simplices with an appropriate notion of connectivity. The proof proceeds verbatim, where we merely replace the simplices $\{T_m\}$ by the convex local patches $\{U_m\}$. This may allow for partitions of $\bar{\Omega}$ with significantly fewer elements, which enables a largely improved estimate of the best Poincaré-Friedrichs constant.

# 5 Review of vector calculus and exterior calculus

We review in this section the Sobolev spaces of vector and exterior calculus with particular emphasis on their behavior under bi-Lipschitz transformations. We refer the reader to Ern and Guermond [24] and Hiptmair [41] for background material on Sobolev vector analysis and to Greub [37] and Lee [48] for exterior algebra and exterior products.

## 5.1 Vector calculus

Let $\Omega \subset \mathbb{R}^3$ be a bounded open set. We recall that $L^p(\Omega)$ denotes the space of scalar-valued $p$-integrable functions defined on $\Omega$ and that $L^p(\Omega) := L^p(\Omega)^n$ stands for vector-valued functions with each component in $L^p(\Omega)$. In the three-dimensional setting, we are particularly interested in the Sobolev vector analysis. The space of scalar-valued $L^p(\Omega)$ functions with weak gradients in $L^p(\Omega)$ is

$$ W^p(\text{grad}, \Omega) := W^{1,p}(\Omega) = \{u \in L^p(\Omega) \mid \text{grad } v \in L^p(\Omega)\}. $$

The space $W^p(\text{curl}, \Omega)$ of vector-valued $L^p(\Omega)$ functions with weak curls in $L^p(\Omega)$ and the space of vector-valued $L^p(\Omega)$ functions with weak divergences in $L^p(\Omega)$ are then written

$$ W^p(\text{curl}, \Omega) = \{u \in L^p(\Omega) \mid \text{curl } u \in L^p(\Omega)\}, $$

$$ W^p(\text{div}, \Omega) = \{u \in L^p(\Omega) \mid \text{div } u \in L^p(\Omega)\}. $$

We are interested in transformations of these Sobolev vector fields from one domain onto another. Suppose that $\Omega, \Omega' \subset \mathbb{R}^3$ are open sets and suppose that $\phi: \Omega \to \Omega'$ is a bi-Lipschitz mapping. We introduce the gradient-, curl-, and divergence-conforming Piola transformations, respectively, as the mappings $\phi^{\text{grad}}: L^p(\Omega') \to L^p(\Omega)$, $\phi^{\text{curl}}: L^p(\Omega') \to L^p(\Omega)$, and $\phi^{\text{div}}: L^p(\Omega') \to L^p(\Omega)$. We also
introduce $\phi^b : L^p(\Omega') \to L^p(\Omega)$. These are defined for any $v \in L^p(\Omega')$ and $w \in L^p(\Omega')$ by setting

$$
\begin{align*}
\phi^{\mathrm{grad}}(v) &= v \circ \phi, \\
\phi^{\mathrm{curl}}(w) &= \mathbf{J}\phi^T(w \circ \phi), \\
\phi^{\mathrm{div}}(w) &= \mathrm{adj}(\mathbf{J}\phi)(w \circ \phi), \\
\phi^b(v) &= \det(\mathbf{J}\phi)(v \circ \phi),
\end{align*}
$$

Here, $\mathbf{J}\phi$ is the Jacobian matrix of $\phi$ (see also [24, Definition 9.8]), and $\mathrm{adj}(\mathbf{J}\phi) := \det(\mathbf{J}\phi)(\mathbf{J}\phi)^{-1}$ denotes taking its adjugate matrix. These transformations are invertible. Bounds on the Lebesgue norms will follow from a more general result below. We use the commutativity relations

$$
\mathrm{grad} \phi^{\mathrm{grad}}(v) = \phi^{\mathrm{curl}}(\mathrm{grad} v), \qquad (39\mathrm{a})
$$

$$
\mathrm{curl} \phi^{\mathrm{curl}}(\mathbf{v}) = \phi^{\mathrm{div}}(\mathrm{curl} \mathbf{v}), \qquad (39\mathrm{b})
$$

$$
\mathrm{div} \phi^{\mathrm{div}}(\mathbf{w}) = \phi^{\mathrm{b}}(\mathrm{div} \mathbf{w}), \qquad (39c)
$$

where $v \in W^p(\mathrm{grad}, \Omega)$, $v \in W^p(\mathrm{curl}, \Omega)$, and $w \in W^p(\mathrm{div}, \Omega)$. We summarize this as a commuting diagram:

$$
\begin{tikzcd}
W^p(\mathrm{grad}, \Omega') \arrow[r, "\mathrm{grad}"] \arrow[d, "\phi^{\mathrm{grad}}"] & W^p(\mathrm{curl}, \Omega') \arrow[r, "\mathrm{curl}"] \arrow[d, "\phi^{\mathrm{curl}}"] & W^p(\mathrm{div}, \Omega') \arrow[r, "\mathrm{div}"] \arrow[d, "\phi^{\mathrm{div}}"] & L^p(\Omega') \arrow[d, "\phi^b"] \\
W^p(\mathrm{grad}, \Omega) \arrow[r, "\mathrm{grad}"] & W^p(\mathrm{curl}, \Omega) \arrow[r, "\mathrm{curl}"] & W^p(\mathrm{div}, \Omega) \arrow[r, "\mathrm{div}"] & L^p(\Omega).
\end{tikzcd}
$$

**Remark 5.1.** The Piola transform goes into the opposite direction of the mapping $\phi : \Omega \to \Omega'$: scalar and vector fields over $\Omega'$ are transformed into scalar and vector fields over $\Omega$. This definition is in accordance with the notion of pullback, which we will review shortly. One advantage of that definition is that it also makes sense whenever the transformation is not bijective. However, the literature also knows the Piola transform in the direction of the original mapping.

5.2 Exterior calculus

We now move the discussion to exterior calculus, beginning with exterior algebra. Let $V$ be a real vector
space. Given an integer $k \ge 0$, we let $\Lambda^k(V)$ denote the space of scalar-valued antisymmetric $k$-linear
forms over $V$. Recall that any $k$-linear scalar-valued form $u$ over $V$ is called antisymmetric if

$$
u(v_{\pi(1)}, v_{\pi(2)}, \dots, v_{\pi(k)}) = \operatorname{sign}(\pi) u(v_1, v_2, \dots, v_k)
$$

for any $v_1, v_2, \dots, v_k \in V$ and any permutation $\pi$ of the indices $\{1, 2, \dots, k\}$. By definition, $\Lambda^1(V)$ is just the dual space of $V$, and $\Lambda^0(V)$ is the space of real numbers. Formally, we define $\Lambda^k(V)$ to be the zero vector space when $k < 0$.

The wedge product (or exterior product) of alternating multilinear forms is a fundamental operation
in exterior algebra (see Chapter 14 in [48]). Given two alternating multilinear forms $u_1 \in \Lambda^k(V)$ and
$u_2 \in \Lambda^l(V)$, their wedge product $u_1 \wedge u_2$ is a member of $\Lambda^{k+l}(V)$ defined by the formula

$$
(u_1 \wedge u_2)(v_1, v_2, \dots, v_{k+l}) = \frac{1}{k!l!} \sum_{\pi} \operatorname{sgn}(\pi) u_1(v_{\pi(1)}, \dots, v_{\pi(k)}) u_2(v_{\pi(k+1)}, \dots, v_{\pi(k+l)}),
$$

for any $v_1, v_2, \dots, v_{k+l} \in V$. Here, the sum runs over all permutations $\pi$ of the index set $\{1, 2, \dots, k+l\}$.
The exterior product is bilinear and associative, and satisfies

$$
u_1 \wedge u_2 = (-1)^{kl} u_2 \wedge u_1, \quad \forall u_1 \in \Lambda^k(V), \quad \forall u_2 \in \Lambda^l(V).
$$

The interior product is in some sense dual to the exterior product. Given $v \in V$ and $u \in \Lambda^k(V)$, we
define the interior product $v \rfloor u \in \Lambda^{k-1}(V)$ via

$$
(v \rfloor u)(v_1, v_2, \dots, v_{k-1}) = u(v, v_1, v_2, \dots, v_{k-1}), \quad \forall v_1, v_2, \dots, v_{k-1} \in V.
$$
We employ the exterior algebra only in the special case $V = \mathbb{R}^n$ of alternating forms over the $n$-dimensional Euclidean space. Here, it is customary to identify $\Lambda^k(V)$ with the space of antisymmetric tensors in $k$ indices. Moreover, this particular setting comes with a canonical basis. We let $\{dx^1, dx^2, \dots, dx^n\}$ be the basis dual to the canonical unit vectors. This is a canonical basis of $\Lambda^1(\mathbb{R}^n)$. To define a canonical basis of $\Lambda^k(\mathbb{R}^n)$, we first introduce $\Sigma(k, n)$, the set of strictly ascending mappings $\sigma: \{1, \dots, k\} \to \{1, \dots, n\}$, where $k, n \in \mathbb{Z}$, and introduce the basic $k$-alternators

$$dx^{\sigma} := dx^{\sigma(1)} \wedge \cdots \wedge dx^{\sigma(k)}, \quad \forall \sigma \in \Sigma(k, n).$$

These define a basis of $\Lambda^k(\mathbb{R}^n)$. Note that $\dim \Lambda^k(\mathbb{R}^n) = \binom{n}{k}$. In particular, $\Lambda^k(\mathbb{R}^n)$ is the zero vector space whenever $k > n$.

We notice that the canonical scalar product on $\mathbb{R}^n$ gives rise to a scalar product on $\Lambda^1(\mathbb{R}^n)$, which induces a scalar product on $\Lambda^k(\mathbb{R}^n)$. The basic $k$-alternators are an orthonormal basis of $\Lambda^k(\mathbb{R}^n)$ with respect to that inner product.

## 5.3 Smooth differential forms

We let $\Omega \subset \mathbb{R}^n$ be any bounded open set. We write $C^\infty\Lambda^k(\Omega)$ for the space of smooth differential $k$-forms over $\Omega$, which is the vector space of smooth mappings from $\Omega$ into $\Lambda^k(\mathbb{R}^n)$. The exterior derivative $d$ is an operator that takes a $k$-form $\omega \in C^\infty\Lambda^k(\Omega)$ to a $(k+1)$-form $d\omega \in C^\infty\Lambda^{k+1}(\Omega)$. Every $k$-form $\omega \in C^\infty\Lambda^k(\Omega)$ can be written

$$\omega = \sum_{\sigma \in \Sigma(k,n)} \omega_{\sigma} dx^{\sigma} = \sum_{\sigma \in \Sigma(k,n)} \omega_{\sigma} dx^{\sigma(1)} \wedge dx^{\sigma(2)} \wedge \cdots \wedge dx^{\sigma(k)}, \quad (40)$$

where $\omega_{\sigma}: \Omega \to \mathbb{R}$ are smooth functions. The exterior derivative $d\omega$ is defined by

$$d\omega = \sum_{\sigma \in \Sigma(k,n)} \sum_{j=1}^{n} \frac{\partial \omega_{\sigma}}{\partial x^{j}} dx^{j} \wedge dx^{\sigma(1)} \wedge dx^{\sigma(2)} \wedge \cdots \wedge dx^{\sigma(k)}.$$

The exterior derivative is linear and nilpotent, which means $d(d\omega) = 0$ for any $\omega \in C^\infty\Lambda^k(\Omega)$. Moreover, it satisfies the Leibniz rule:

$$d(\omega \wedge \eta) = d\omega \wedge \eta + (-1)^k \omega \wedge d\eta, \quad \forall \omega \in C^\infty\Lambda^k(\Omega), \quad \forall \eta \in C^\infty\Lambda^l(\Omega).$$

The integral of a differential $n$-form is uniquely defined via

$$\int_{\Omega} \omega dx^1 \wedge \cdots \wedge dx^n = \int_{\Omega} \omega(x) dx.$$

**Remark 5.2.** In three dimensions, the calculus of differential forms is in correspondence with classical vector calculus. This is expressed formally as the commuting diagram

$$\begin{tikzcd}
C^\infty\Lambda^0(\Omega) \arrow[r, "d"] & C^\infty\Lambda^1(\Omega) \arrow[r, "d"] & C^\infty\Lambda^2(\Omega) \arrow[r, "d"] & C^\infty\Lambda^3(\Omega) \\
\bigwedge \omega^0 \arrow[r, "\text{grad}", "\omega^1"] & C^\infty(\Omega)^3 \arrow[r, "\text{curl}", "\omega^2"] & C^\infty(\Omega)^3 \arrow[r, "\text{div}", "\omega^3"] & C^\infty(\Omega)
\end{tikzcd}$$

where $\omega^0$ and $\omega^3$ are the identity mappings and where

$$\begin{align*}
\omega^1 (u_1 dx^1 + u_2 dx^2 + u_3 dx^3) &= (u_1, u_2, u_3), \\
\omega^2 (u_{12} dx^1 \wedge dx^2 + u_{13} dx^1 \wedge dx^3 + u_{23} dx^2 \wedge dx^3) &= (u_{23}, -u_{13}, u_{12}).
\end{align*}$$

In two dimensions, the calculus of differential forms can be translated into 2D vector calculus in two different ways. To the authors' best knowledge, neither convention is dominant over the other in the
literature. We summarize the situation in the following commuting diagram:

$$
\begin{tikzcd}
C^\infty(\Omega) \arrow[r, "\mathrm{curl}"] \arrow[d, "\varkappa^0"] & C^\infty(\Omega)^2 \arrow[r, "\mathrm{div}"] \arrow[d, "\varkappa^1"] & C^\infty(\Omega) \\
C^\infty\Lambda^0(\Omega) \arrow[r, "d"] & C^\infty\Lambda^1(\Omega) \arrow[r, "d"] & C^\infty\Lambda^2(\Omega) \\
C^\infty(\Omega) \arrow[r, "\mathrm{grad}"] \arrow[d, "\varkappa^0"] & C^\infty(\Omega)^2 \arrow[r, "\mathrm{rot}"] \arrow[d, "\varkappa^1"] & C^\infty(\Omega).
\end{tikzcd}
$$

Here, $\bar{\omega}^1(u_1dx^1 + u_2dx^2) = (u_1, u_2)$ is the lower middle isomorphism. We introduce the rotation operator $J(x, y) = (y, -x)$ and define $\varkappa^1 = J\bar{\omega}^1$ and $\mathrm{rot} = \mathrm{div} \, J$. The other vertical arrows are the identity. The utility of exterior calculus is that the operators of vector calculus can be translated into a common framework that does not depend on the dimension.

5.4 Sobolev spaces of differential forms

Let us now turn our attention to Sobolev spaces of differential forms. Since the exterior product space
$\Lambda^k(\mathbb{R}^n)$ carries a norm, induced from the Euclidean norm on $\mathbb{R}^n$, there are pointwise norms of differential
$k$-forms. We let $L^p\Lambda^k(\Omega)$ be the space of differential $k$-forms over $\Omega$ with locally integrable coefficients
such that its pointwise norm is $p$-integrable. The exterior derivative is defined in the sense of distributions
and we introduce

$$
W^p \Lambda^k (\Omega) := \{ u \in L^p \Lambda^k (\Omega) \mid du \in L^p \Lambda^{k+1} (\Omega) \}.
$$

We observe that $u \in L^p \Lambda^k(\Omega)$ has weak exterior derivative $f \in L^p \Lambda^{k+1}(\Omega)$ if and only if for all $v \in C_c^\infty(\Lambda^{n-k-1}(\Omega))$ we have the integration-by-parts formula

$$
\int_{\Omega} dv \wedge u = (-1)^{k(n-k)+1} \int_{\Omega} v \wedge f.
$$

Lastly, we are also interested in differential forms whose trace vanishes along a part of the boundary.
Suppose that $\Gamma \subseteq \partial\Omega$ is a relatively open subset of the boundary. We say that $u \in W^p\Lambda^k(\Omega)$ has
vanishing trace along $\Gamma$ if for all $x \in \Gamma$ there exists $r > 0$ such that for all $v \in C_c^\infty(\Lambda^{n-k-1}(\mathbb{R}^n))$ whose
support lies in the open ball $B_r(x)$ we have the integration-by-parts formula

$$
\int_{B_r(x)} dv \wedge \tilde{u} = (-1)^{k(n-k)+1} \int_{B_r(x)} v \wedge \tilde{du},
$$

where $\tilde{u} : \mathbb{R}^n \to \Lambda^k(\mathbb{R}^n)$ and $\tilde{du} : \mathbb{R}^n \to \Lambda^{k+1}(\mathbb{R}^n)$ are the extensions of $u$ and $du$ by zero. If that
condition is satisfied, we also write

$$
\mathrm{tr}_{\Gamma} u = 0.
$$

Accordingly, we write tr$_{\Gamma} u = \mathrm{tr}_{\Gamma} u'$ for tr$_{\Gamma}(u - u') = 0$ whenever $u, u' \in W^p \Lambda^k(\Omega)$. Lastly, we introduce
the closed subspaces

$$
W_0^p \Lambda^k(\Omega) := \{ u \in W^p \Lambda^k(\Omega) \mid \mathrm{tr}_{\partial\Omega} u = 0 \}.
$$

We know that $u \in W_0^p \Lambda^k(\Omega)$ if and only if its extension by zero $\tilde{u} : \mathbb{R}^n \to \Lambda^k(\mathbb{R}^n)$ is a member of
$\tilde{u} \in W^p \Lambda^k(\mathbb{R}^n)$. Moreover, $dW_0^p \Lambda^k(\Omega) \subseteq W_0^{p,k+1}(\Omega)$. We also observe that $W^p \Lambda^n(\Omega) =
L^p(\Omega)$. We use the abbreviation $W_0^{1,p}(\Omega) := W^p \Lambda^0(\Omega)$.

5.5 Transformations by bi-Lipschitz mappings

We are interested in transformations of Sobolev tensor fields from one domain onto another. Suppose that $\Omega, \Omega' \subset \mathbb{R}^n$ are open sets and suppose that $\phi : \Omega \to \Omega'$ is a bi-Lipschitz mapping. The pullback of $u \in L^p \Lambda^k(\Omega')$ along $\phi$ is the (measurable) differential form

$$
\phi^* u|_x(v_1, v_2, \dots, v_k) := u|_{\phi(x)}(\mathbf{J}\phi|_x \cdot v_1, \mathbf{J}\phi|_x \cdot v_2, \dots, \mathbf{J}\phi|_x \cdot v_k) \quad (41)
$$
for any $v_1, v_2, \dots, v_k \in \mathbb{R}^n$ and any $x \in \Omega$. One can show that $\phi^*u \in L^p\Lambda^k(\Omega)$ and the following estimates.

**Proposition 5.3.** Let $\phi : \Omega \to \Omega'$ be a bi-Lipschitz mapping between open sets $\Omega, \Omega' \subseteq \mathbb{R}^n$. Let $p \in [1, \infty]$ and $u \in L^p\Lambda^k(\Omega')$. Then $\phi^*u \in L^p\Lambda^k(\Omega)$ and

$$ \|\phi^*u\|_{L^p\Lambda^k(\Omega)} \le \|\mathbf{J}\phi\|_{L^\infty(\Omega)}^k \det \mathbf{J}\phi^{-1} \|u\|_{L^p\Lambda^k(\Omega')}. \quad (42) $$

If $\phi$ is affine and $\sigma_1 \ge \sigma_2 \ge \dots \ge \sigma_n$ are the singular values of $\mathbf{J}\phi$, then

$$ \|\phi^* u\|_{L^p \Lambda^k(\Omega)} \le \sigma_1 \sigma_2 \cdots \sigma_k \|\det \mathbf{J} \phi^{-1}\|_{L^\infty(\Omega')}^{\frac{1}{p}} \|u\|_{L^p \Lambda^k(\Omega')}. \quad (43) $$

Moreover, if $u \in W^p\Lambda^k(\Omega')$, then

$$ \phi^* u \in W^p \Lambda^k(\Omega), \quad d\phi^* u = \phi^* du. \quad (44) $$

*Proof.* See [50] and Corollary 6 in [64]. □

## 5.6 Some approximation properties

We review a few approximation properties. Let $\mathbf{m}$ be a non-negative scalar function whose integral equals one and whose support lies in the unit ball around the origin. Define $\mathbf{m}_\epsilon(x) := \epsilon^{-n}\mathbf{m}(x/\epsilon)$. We collect several approximation results that involve convolution with a mollifier. In what follows, convolutions of functions defined over domains tacitly assume that these functions have been extended by zero to all of Euclidean space.

**Lemma 5.4.** Let $\Omega \subseteq \mathbb{R}^n$ be a bounded open set and let $1 \le p < \infty$. If $u \in L^p(\Omega)$, then the convolution $\mathbf{m}_\epsilon \star u \to u$ in $L^p(\Omega)$ as $\epsilon \to 0$.

**Lemma 5.5.** Let $\Omega \subseteq \mathbb{R}^n$ be a bounded open set and let $1 \le p < \infty$. Smooth forms are dense in $W^p\Lambda^k(\Omega)$. If $\Omega$ is convex, then $C_c^\infty\Lambda^k(\Omega)$ is dense in $W_0^p\Lambda^k(\Omega)$.

*Proof.* We notice $\mathbf{m}_\epsilon \star u \in C^\infty\Lambda^k(\mathbb{R}^n)$. By the dominated convergence theorem and because $u$ has a weak derivative, for any $v \in C_c^\infty\Lambda^{n-k-1}(\Omega)$

$$ \begin{aligned} \int_{\mathbb{R}^n} (\mathbf{m}_\epsilon \star u) \wedge dv &= \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \mathbf{m}(x-y) \wedge u(y) \wedge d_x v(x) \, dx \, dy \\ &= \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} d_x \mathbf{m}(x-y) \wedge u(y) \wedge v(x) \, dx \, dy \\ &= - \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} d_y \mathbf{m}(x-y) \wedge u(y) \wedge v(x) \, dx \, dy \\ &= \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \mathbf{m}(x-y) \wedge d_y u(y) \wedge v(x) \, dx \, dy = \int_{\mathbb{R}^n} (\mathbf{m}_\epsilon \star du) \wedge v. \end{aligned} $$

Hence, $C^\infty\Lambda^k(\Omega)$ is dense in $W^p\Lambda^k(\Omega)$.

Next, suppose that $\Omega$ is convex. Without loss of generality, $0 \in \Omega$. Let $u \in W_0^p\Lambda^k(\Omega)$ and extend $u$ trivially onto $\mathbb{R}^n$. Define $\varphi_t(x) = tx$ for $t > 1$. Then $\varphi_t^*u \in W_0^p\Lambda^k(t^{-1}\Omega) \subseteq W_0^p\Lambda^k(\Omega)$ and $\varphi_t^*u$ converges to $u$ as $t$ decreases towards 1. Given any $t > 1$, by taking the convolution with $\mathbf{m}_\epsilon$ for $\epsilon > 0$ small enough, we approximate $\varphi_t^*u$ through members of $C_c^\infty\Lambda^k(\Omega)$. The desired result follows. □

## 6 Regularized potentials over bounded convex sets

We now develop bounds for Poincaré-Friedrichs constants for the exterior derivative over convex domains. Here, we consider two special cases: either the $L^p$ de Rham complex without boundary conditions, or the $L^p$ de Rham with full boundary conditions. The corresponding linear potentials are known as the regularized Poincaré and regularized Bogovskiï potentials in the literature. We build upon the discussion spearheaded by Costabel and McIntosh [22], who analyze them as pseudo-differential operators over
domains star-shaped with respect to a ball. In comparison to their extensive work, our discussion is more modest: we study potentials merely over convex sets, and we are only interested in their operator norms between Lebesgue spaces. However, our goal is explicit bounds for the operator norms, giving the Poincaré-Friedrichs constants.

In the remainder of this section, $\Omega \subseteq \mathbb{R}^n$ is a bounded convex open set with diameter $\delta(\Omega) > 0$.

## 6.1 Regularized Poincaré and Bogovskiǐ operators

We begin by introducing the Costabel-McIntosh kernel. For any $k \in \{0, \dots, n\}$, we define the kernel $G_k : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ by

$$G_k(x, y) = \int_1^\infty (t-1)^{n-k} t^{k-1} \mathrm{vol}(\Omega)^{-1} \chi_\Omega (y+t(x-y)) \, dt, \quad (45)$$

where $\chi_\Omega : \Omega \to \{0, 1\}$ denotes the characteristic function of the domain $\Omega$. Given a differential form $u \in C_c^\infty(\mathbb{R}^n, \Lambda^k)$, where $1 \le k \le n$, we then define the integral operators

$$\mathcal{P}_k u(x) = \int_{\Omega} G_{n-k+1}(y,x) (x-y) \rfloor u(y) \, dy,$$

$$\mathcal{B}_k u(x) = \int_{\Omega} G_k(x, y) (x - y) \rfloor u(y) \, dy.$$

We call $\mathcal{P}_k$ the Poincaré operator and $\mathcal{B}_k$ the Bogovskiǐ operator.

We show that the integrals in the definition of $\mathcal{P}_k$ and $\mathcal{B}_k$ actually converge. In order to analyze the properties of the potentials, we first rewrite the Costabel-McIntosh kernel $G_k$. Letting $x, y \in \mathbb{R}^n$ with $x \neq y$, we find

$$\begin{align*}
G_k(x, y) &= \int_0^\infty t^{n-k} (t+1)^{k-1} \mathrm{vol}(\Omega)^{-1} \chi_\Omega (x+t(x-y)) \, dt \\
&= \int_0^\infty \sum_{\ell=0}^{k-1} \binom{k-1}{\ell} t^{n-k+\ell} \mathrm{vol}(\Omega)^{-1} \chi_\Omega (x+t(x-y)) \, dt \\
&= \int_0^\infty \sum_{\ell=0}^{k-1} \binom{k-1}{k-1-\ell} t^{n-k+k-1-\ell} \mathrm{vol}(\Omega)^{-1} \chi_\Omega (x+t(x-y)) \, dt \\
&= \int_0^\infty \sum_{\ell=0}^{k-1} \binom{k-1}{\ell} t^{n-\ell-1} \mathrm{vol}(\Omega)^{-1} \chi_\Omega (x+t(x-y)) \, dt \\
&= \sum_{\ell=0}^{k-1} \binom{k-1}{\ell} \int_0^\infty t^{n-\ell-1} \mathrm{vol}(\Omega)^{-1} \chi_\Omega (x+t(x-y)) \, dt \\
&= \mathrm{vol}(\Omega)^{-1} \sum_{\ell=0}^{k-1} \binom{k-1}{\ell} |x-y|^{\ell-n} \int_0^\infty r^{n-\ell-1} \chi_\Omega \left(x+r \frac{x-y}{|x-y|}\right) \, dr.
\end{align*}$$

If $x \in \Omega$ and $x \neq y$, then we can restrict the inner integrals to the range $0 \le r \le \delta(\Omega)$, which gives

$$\begin{align*}
G_k(x, y) &= \mathrm{vol}(\Omega)^{-1} \sum_{\ell=0}^{k-1} \binom{k-1}{\ell} |x-y|^{\ell-n} \int_0^{\delta(\Omega)} r^{n-\ell-1} \, dr \\
&= \mathrm{vol}(\Omega)^{-1} \sum_{\ell=0}^{k-1} \binom{k-1}{\ell} |x-y|^{\ell-n} \frac{\delta(\Omega)^{n-\ell}}{n-\ell}.
\end{align*}$$

We are now in a position to show that the potentials are bounded with respect to Lebesgue norms.

## 6.2 An operator norm bound with respect to the Lebesgue norm: the Poincaré case

We begin with the Poincaré operator. Let $B_\delta(\Omega)(0)$ be the $n$-dimensional ball centered at the origin. Suppose that $u \in L^\infty \Lambda^k(\Omega)$ with $1 \le k \le n$. We estimate $\mathcal{P}_k u(x)$ pointwise for any $x \in \Omega$ by the result
of a convolution of a locally integrable function with $u$:

$$ \begin{aligned} |\mathcal{P}_k u(x)| &= \left| \int_{\Omega} \mathcal{G}_{n-k+1}(x,y) (x-y) u(y) dy \right| \\ &\le \int_{\Omega} \mathrm{vol}(\Omega)^{-1} \sum_{\ell=0}^{n-k} \binom{n-k}{\ell} \frac{\delta(\Omega)^{n-\ell}}{n-\ell} |x-y|^{\ell+1-n} \chi_{B_{\delta(\Omega)}(0)}(x-y) |u(y)| dy. \end{aligned} $$

We recall the radial integrals

$$ \begin{aligned} \int_{B_{\delta(\Omega)}(0)} |z|^{\ell+1-n} dz &= \mathrm{vol}_{n-1}(S_1) \int_0^{\delta(\Omega)} r^{\ell+1-n} r^{n-1} dr \\ &= \mathrm{vol}_{n-1}(S_1) \int_0^{\delta(\Omega)} r^{\ell} dr = \mathrm{vol}_{n-1}(S_1) \frac{\delta(\Omega)^{\ell+1}}{\ell+1}, \end{aligned} $$

where $S_1 \subseteq \mathbb{R}^n$ stands for the unit sphere of dimension $n-1$. One computes

$$ \begin{aligned} & \int_{\mathbb{R}^n} \sum_{\ell=0}^{n-k} \binom{n-k}{\ell} \frac{\delta(\Omega)^{n-\ell}}{n-\ell} \chi_{B_{\delta(\Omega)}(0)}(z) |z|^{\ell+1-n} dz \\ &= \sum_{\ell=0}^{n-k} \binom{n-k}{\ell} \frac{\delta(\Omega)^{n-\ell}}{n-\ell} \int_{B_{\delta(\Omega)}(0)} |z|^{\ell+1-n} dz \\ &= \mathrm{vol}_{n-1}(S_1) \sum_{\ell=0}^{n-k} \binom{n-k}{\ell} \frac{\delta(\Omega)^{n-\ell}}{n-\ell} \frac{\delta(\Omega)^{\ell+1}}{\ell+1} = \mathrm{vol}_{n-1}(S_1) \delta(\Omega)^{n+1} \underbrace{\sum_{\ell=0}^{n-k} \frac{\binom{n-k}{\ell}}{(n-\ell)(\ell+1)}}_{=:C_P(n,k) \le 2^{n-k}}. \end{aligned} $$

Here we introduce the numerical constant

$$ C_P(n, k) := \sum_{\ell=0}^{n-k} \frac{\binom{n-k}{\ell}}{(n-\ell)(\ell+1)}, \qquad (46) $$

which depends only on $n$ and $k$ and which is bounded by $2^{n-k}$.

In particular, the integral $\mathcal{P}_k u(x)$ is absolutely convergent for any choice of $x \in \Omega$ and its magnitude is pointwise dominated by the convolution of $|u|$ against an integrable function. Young's convolution inequality now implies:

$$ \begin{aligned} \|\mathcal{P}_k u\|_{L^p(\Omega)} &\le \mathrm{vol}_{n-1}(S_1) C_P(n, k) \frac{\delta(\Omega)^n}{\mathrm{vol}(\Omega)} \delta(\Omega) \|u\|_{L^p(\Omega)} \\ &\le n C_P(n, k) \frac{\mathrm{vol}(B_{\delta(\Omega)}(0))}{\mathrm{vol}(\Omega)} \delta(\Omega) \|u\|_{L^p(\Omega)}. \end{aligned} $$

We have assumed so far that $u \in L^\infty \Lambda^k(\Omega)$. Since that space is dense in the Lebesgue spaces, a density argument establishes the following: for any $1 \le p \le \infty$ we have a bounded linear operator

$$ \mathcal{P}_k : L^p \Lambda^k(\Omega) \to L^p \Lambda^{k-1}(\mathbb{R}^n). $$

**Remark 6.1.** As already explained in the original work of Costabel and McIntosh [22], the operators $\mathcal{P}_k$ preserve polynomial differential forms.

## 6.3 An operator norm bound with respect to the Lebesgue norm: the Bogovskiï case

We analyze the Bogovskiï potential operator by similar means. Suppose that $u \in L^\infty \Lambda^k(\mathbb{R}^n)$ with $\mathrm{supp} u \subseteq \bar{\Omega}$ and that $x \in \mathbb{R}^n$. First, if $x \notin \Omega$, then the convexity of $\Omega$ implies that $y + t(x - y) \notin \Omega$ for
all $t > 1$. Hence, $G_k(x, y) = 0$ and therefore $B_k u(x) = 0$ in that case. Consider now the case $x \in \bar{\Omega}$. We estimate $B_k u(x)$ pointwise by

$$ \begin{aligned} |B_k u(x)| &= \left| \int_{\Omega} G_k(x, y) (x-y) \rfloor u(y) dy \right| \\ &\le \int_{\Omega} \mathrm{vol}(\Omega)^{-1} \sum_{\ell=0}^{k-1} \binom{k-1}{\ell} \frac{\delta(\Omega)^{n-\ell}}{n-\ell} |x-y|^{\ell+1-n} |u(y)| dy \\ &\le \int_{\mathbb{R}^n} \mathrm{vol}(\Omega)^{-1} \sum_{\ell=0}^{k-1} \binom{k-1}{\ell} \frac{\delta(\Omega)^{n-\ell}}{n-\ell} \chi_{B_{\delta(\Omega)}(0)}(x-y) |x-y|^{\ell+1-n} |u(y)| dy. \end{aligned} $$

Using once more the radial integrals discussed above, we compute

$$ \begin{aligned} & \int_{\mathbb{R}^n} \sum_{\ell=0}^{k-1} \binom{k-1}{\ell} \frac{\delta(\Omega)^{n-\ell}}{n-\ell} \chi_{B_{\delta(\Omega)}(0)}(z) |z|^{\ell+1-n} dz \\ &= \sum_{\ell=0}^{k-1} \binom{k-1}{\ell} \frac{\delta(\Omega)^{n-\ell}}{n-\ell} \int_{B_{\delta(\Omega)}(0)} |z|^{\ell+1-n} dz \\ &= \mathrm{vol}_{n-1}(S_1) \sum_{\ell=0}^{k-1} \binom{k-1}{\ell} \frac{\delta(\Omega)^{n-\ell}}{n-\ell} \frac{\delta(\Omega)^{\ell+1}}{\ell+1} = \mathrm{vol}_{n-1}(S_1) \delta(\Omega)^{n+1} \underbrace{\sum_{\ell=0}^{k-1} \frac{\binom{k-1}{\ell}}{(n-\ell)(\ell+1)}}_{=:C_B(n,k) \le 2^{k-1}}. \end{aligned} $$

Here we introduce the numerical constant

$$ C_B(n, k) := \sum_{\ell=0}^{k-1} \frac{\binom{k-1}{\ell}}{(n-\ell)(\ell+1)}, \qquad (47) $$

which depends only on $n$ and $k$ and which is bounded by $2^{k-1}$. Similar as above, the integral $B_k u(x)$ is absolutely convergent for any choice of $x \in \mathbb{R}^n$ and its magnitude is pointwise dominated by the convolution of $|u|$ against an integrable function. Young's convolution inequality now implies:

$$ \begin{aligned} \|B_k u\|_{L^p(\Omega)} &\le \mathrm{vol}_{n-1}(S_1) C_B(n, k) \frac{\delta(\Omega)^n}{\mathrm{vol}(\Omega)} \delta(\Omega) \|u\|_{L^p(\Omega)} \\ &\le n C_B(n, k) \frac{\mathrm{vol}(B_{\delta(\Omega)}(0))}{\mathrm{vol}(\Omega)} \delta(\Omega) \|u\|_{L^p(\Omega)}. \end{aligned} $$

We have assumed so far that $u$ is essentially bounded. Since that space is dense in the Lebesgue spaces, a density argument yields: for any $1 \le p \le \infty$ we have a bounded linear operator

$$ B_k : L^p \Lambda^k(\Omega) \to L^p \Lambda^{k-1}(\mathbb{R}^n). $$

Moreover, $\mathrm{supp} \, B_k u \subseteq \bar{\Omega}$, that is, the reconstructed potential has support contained within $\bar{\Omega}$.

## 6.4 Rewriting the potentials operators

More properties of these operators become apparent after a change of variables. We write down the full definition of these operators and perform two substitutions. For the Poincaré operator, we substitute $a = x + t(y - x)$, and then we substitute $s = (t - 1)/t$, leading to

$$ \begin{aligned} \mathcal{P}_k u(x) &= \mathrm{vol}(\Omega)^{-1} \int_{\Omega} \int_{1}^{\infty} (t-1)^{k-1} t^{n-k} \chi_{\Omega} (x+t(y-x)) (x-y) \rfloor u(y) dt dy \\ &= \mathrm{vol}(\Omega)^{-1} \int_{\mathbb{R}^n} \chi_{\Omega}(a) (x-a) \rfloor \int_{0}^{1} t^{k-1} u(a+t(x-a)) dt da. \end{aligned} $$
For the Bogovskiï operator, we first substitute $a = y + t(x - y)$, and then we substitute $s = t/(t - 1)$, leading to

$$ 
\begin{aligned}
\mathcal{B}_k u(x) &= \mathrm{vol}(\Omega)^{-1} \int_{\Omega} \int_{1}^{\infty} (t-1)^{n-k} t^{k-1} \chi_{\Omega} (y+t(x-y))(x-y) \rfloor u(y) \, dt \, dy \\
&= -\mathrm{vol}(\Omega)^{-1} \int_{\mathbb{R}^n} \chi_{\Omega}(a)(x-a) \rfloor \int_{1}^{\infty} t^{k-1} u(a+t(x-a)) \, dt \, da.
\end{aligned}
$$

Given $a \in \Omega$, we introduce the potentials

$$ \mathcal{P}_{k,a}u(x) := (x-a)\rfloor \int_{0}^{1} t^{k-1}u(a+t(x-a)) \, dt, $$

$$ \mathcal{B}_{k,a}u(x) := -(x-a)\rfloor \int_{1}^{\infty} t^{k-1}u(a+t(x-a)) \, dt. $$

By definition,

$$ \mathcal{P}_k u(x) = \mathrm{vol}(\Omega)^{-1} \int_{\Omega} \mathcal{P}_{k,a} u(x) \, da, \quad \mathcal{B}_k u(x) = \mathrm{vol}(\Omega)^{-1} \int_{\Omega} \mathcal{B}_{k,a} u(x) \, da. $$

## 6.5 Interaction of potentials with the exterior derivative

We study the interaction of the Poincaré and Bogovskiï operators with the exterior derivative in more detail. The main arguments are well-known and establish the exactness of several de Rham complexes. We recapitulate these arguments since our variants of the regularized potential operators are not yet included in the published literature.

We make use of the following notation [51]: for any mapping $\sigma \in \Sigma(k, n)$, we let $[\sigma] := \{\sigma(1), \dots, \sigma(k)\}$ be its image. When $p \in [\sigma]$, then the member of $\Sigma(k-1, n)$ with image $[\sigma] \setminus \{p\}$ is written $\sigma - p$.

Suppose that $u \in C^{\infty}\Lambda^k(\mathbb{R}^n)$. We rewrite the Poincaré potential,

$$ 
\begin{aligned}
\mathcal{P}_{k,a}u(x) &= (x-a)\rfloor \int_{0}^{1} t^{k-1}u(a+t(x-a)) \, dt \\
&= (x-a)\rfloor \sum_{\sigma \in \Sigma(k,n)} \int_{0}^{1} t^{k-1}u_{\sigma}(a+t(x-a)) \, dx^{\sigma} \, dt \\
&= \sum_{\sigma \in \Sigma(k,n)} \sum_{i=1}^{k} \int_{0}^{1} t^{k-1}u_{\sigma}(a+t(x-a))(-1)^{i-1}(x-a)_{\sigma(i)} \, dx^{\sigma-\sigma(i)} \, dt,
\end{aligned}
$$

and compute its exterior derivative:

$$ 
\begin{aligned}
d\mathcal{P}_{k,a}u(x) = & \sum_{\sigma \in \Sigma(k,n)} \int_{0}^{1} kt^{k-1}u_{\sigma}(a+t(x-a)) \, dx^{\sigma} \, dt \\
& + \sum_{\substack{\sigma \in \Sigma(k,n), 1 \le i \le k \\ 1 \le j \le n, j \notin [\sigma-\sigma(i)]}} \int_{0}^{1} t^{k} \frac{\partial u_{\sigma}}{\partial x_{j}}(a+t(x-a))(-1)^{i-1}(x-a)_{\sigma(i)} \, dx^{j} \wedge dx^{\sigma-\sigma(i)} \, dt.
\end{aligned}
$$

We write the exterior derivative of $u$ as

$$ du(x) = \sum_{\sigma \in \Sigma(k,n)} \sum_{j=1}^{n} \frac{\partial u}{\partial x_j}(x) \, dx^j \wedge dx^\sigma = \sum_{\substack{\sigma \in \Sigma(k,n) \\ 1 \le j \le n, j \notin [\sigma]}} \frac{\partial u}{\partial x_j}(x) \, dx^j \wedge dx^\sigma, \quad (48) $$
and apply the Poincaré potential operator to this result, which gives

$$
\begin{align*}
\mathcal{P}_{k+1,a} du(x) &= (x-a) \sum_{\substack{\sigma \in \Sigma(k,n) \\ 1 \le j \le n, j \notin [\sigma]}} \int_0^1 t^k \frac{\partial u_\sigma}{\partial x_j} (a+t(x-a)) \, dt \, dx^j \wedge dx^\sigma \\
&= \sum_{\substack{\sigma \in \Sigma(k,n) \\ 1 \le j \le n, j \notin [\sigma]}} \int_0^1 t^k \frac{\partial u_\sigma}{\partial x_j} (a+t(x-a)) \, dt (x-a)_j \, dx^\sigma \\
&\quad - \sum_{\substack{\sigma \in \Sigma(k,n) \\ 1 \le j \le n, j \notin [\sigma] \\ 1 \le i \le k}} (-1)^{i-1} \int_0^1 t^k \frac{\partial u_\sigma}{\partial x_j} (a+t(x-a)) \, dt (x-a)_{\sigma(i)} \, dx^j \wedge dx^{\sigma-\sigma(i)}.
\end{align*}
$$

Next, we add the exterior derivative of the potential and the potential of the exterior derivative. Taking
into account cancellations, this gives the identity

$$
\begin{align*}
& d\mathcal{P}_{k,a}u(x) + \mathcal{P}_{k+1,a}du(x) \\
&= \sum_{\sigma \in \Sigma(k,n)} \int_0^1 kt^{k-1}u_\sigma (a+t(x-a)) \, dx^{\sigma-\sigma(i)} \, dt \\
&\quad + \sum_{\substack{\sigma \in \Sigma(k,n), 1 \le i \le k \\ 1 \le j \le n, j \notin [\sigma-\sigma(i)]}} \int_0^1 t^k \frac{\partial u_\sigma}{\partial x_j} (a+t(x-a)) (-1)^{i-1} (x-a)_{\sigma(i)} \, dx^j \wedge dx^{\sigma-\sigma(i)} \, dt \\
&\quad + \sum_{\substack{\sigma \in \Sigma(k,n) \\ 1 \le j \le n, j \notin [\sigma]}} \int_0^1 t^k \frac{\partial u_\sigma}{\partial x_j} (a+t(x-a)) \, dt (x-a)_j \, dx^\sigma \\
&\quad - \sum_{\substack{\sigma \in \Sigma(k,n) \\ 1 \le j \le n, j \notin [\sigma] \\ 1 \le i \le k}} (-1)^{i-1} \int_0^1 t^k \frac{\partial u_\sigma}{\partial x_j} (a+t(x-a)) \, dt (x-a)_{\sigma(i)} \, dx^j \wedge dx^{\sigma-\sigma(i)} \\
&= \sum_{\sigma \in \Sigma(k,n)} \int_0^1 kt^{k-1}u_\sigma (a+t(x-a)) \, dx^{\sigma-\sigma(i)} \, dt \\
&\quad + \sum_{\substack{\sigma \in \Sigma(k,n) \\ 1 \le j \le n, j \notin [\sigma]}} \int_0^1 t^k \frac{\partial u_\sigma}{\partial x_j} (a+t(x-a)) (x-a)_j \, dt \, dx^\sigma \\
&\quad + \sum_{\substack{\sigma \in \Sigma(k,n), 1 \le i \le k \\ 1 \le j \le n, j \notin [\sigma]}} \int_0^1 t^k \frac{\partial u_\sigma}{\partial x_{\sigma(i)}} (a+t(x-a)) (-1)^{i-1} (x-a)_{\sigma(i)} \, dx^{\sigma(i)} \wedge dx^{\sigma-\sigma(i)} \, dt \\
&= \sum_{\sigma \in \Sigma(k,n)} \int_0^1 \frac{\partial}{\partial t} (t^k u_\sigma (a+t(x-a))) \, dt \, dx^\sigma = \sum_{\sigma \in \Sigma(k,n)} (u_\sigma(x) - 0^k u_\sigma(a)) \, dx^\sigma.
\end{align*}
$$

We conclude that,

$$
\begin{align*}
u(x) &= \mathcal{P}_{1,a}du(x) - u(a), && k=0, \\
u(x) &= d\mathcal{P}_{k,a}u(x) + \mathcal{P}_{k+1,a}du(x), && 1 \le k \le n.
\end{align*}
$$

In summary, after taking the average over $a \in \Omega$:

$$
u(x) = \mathcal{P}_1 du(x) - \text{vol}(\Omega)^{-1} \int_{\Omega} u(a) \, da, \quad k=0, \tag{49}
$$

$$
u(x) = d\mathcal{P}_k u(x) + \mathcal{P}_{k+1} du(x), \quad 1 \le k \le n. \tag{50}
$$
Even though the discussion for the Bogovskiǐ operator is large analogous, some modifications are needed. Suppose that $u \in C^\infty\Lambda^k(\mathbb{R}^n)$ with supp $u \subseteq \bar{\Omega}$. We rewrite the Bogovskiǐ potential,

$$ 
\begin{aligned}
-\mathcal{B}_{k,a}u(x) &= (x-a)\int_1^\infty t^{k-1}u(a+t(x-a))dt \\
&= (x-a)\sum_{\sigma \in \Sigma(k,n)} \int_1^\infty t^{k-1}u_\sigma(a+t(x-a))dx^\sigma dt \\
&= \sum_{\sigma \in \Sigma(k,n)} \sum_{i=1}^k \int_1^\infty t^{k-1}u_\sigma(a+t(x-a))(-1)^{i-1}(x-a)_{\sigma(i)}dx^{\sigma-\sigma(i)}dt.
\end{aligned}
$$

We want to take its exterior derivative, but we can generally only do that in the distributional sense. Away from the pivot point $a$, the form $\mathcal{B}_{k,a}u(x)$ is differentiable in $x$, and so we compute its exterior derivative over $\Omega \setminus \{a\}$:

$$ 
\begin{aligned}
-d\mathcal{B}_{k,a}u(x) = & \sum_{\sigma \in \Sigma(k,n)} \int_1^\infty kt^{k-1}u_\sigma(a+t(x-a))dx^\sigma dt \\
& + \sum_{\substack{\sigma \in \Sigma(k,n), 1 \le i \le k \\ 1 \le j \le n, j \notin [\sigma-\sigma(i)]}} \int_1^\infty t^k \frac{\partial u_\sigma}{\partial x_j}(a+t(x-a))(-1)^{i-1}(x-a)_{\sigma(i)}dx^j \wedge dx^{\sigma-\sigma(i)}dt.
\end{aligned}
$$

The derivative of $\mathcal{B}_{k,a}u$ over the whole domain, which is what we need, can only be taken in the sense of distributions. Let $\phi \in C^\infty\Lambda^{n-k}(\Omega)$ be smooth and compactly supported over $\Omega$. We let $\epsilon > 0$ and calculate

$$ (-1)^{n(k-1)} \int_{\Omega \setminus B_{\epsilon}(a)} \mathcal{B}_{k,a}u(x) \wedge d\phi = \int_{S_{\epsilon}(a)} \mathrm{tr}_{S_{\epsilon}(a)} \mathcal{B}_{k,a}u(x) \wedge \phi - \int_{\Omega \setminus B_{\epsilon}(a)} d\mathcal{B}_{k,a}u(x) \wedge \phi, $$

where $\mathrm{tr}_{S_{\epsilon}(a)}$ denotes the trace onto the sphere $S_{\epsilon}(a)$. In the limit as $\epsilon$ goes to zero, the two integrals over $\Omega \setminus B_{\epsilon}(a)$ in the above equation converge to the integrals of the respective integrands over $\Omega \setminus \{a\}$. To understand the derivative of $\mathcal{B}_{k,a}u$ over the domain $\Omega$, we study the remaining surface integral. We apply several substitutions:

$$ 
\begin{aligned}
& \int_{S_{\epsilon}(a)} \mathrm{tr}_{S_{\epsilon}(a)} \mathcal{B}_{k,a} u(x) \wedge \phi(x) \\
&= \epsilon^{n-1} \int_{S_1(a)} \mathrm{tr}_{S_1(a)} \mathcal{B}_{k,a} u(\epsilon x + a - \epsilon a) \wedge \phi(\epsilon x + a - \epsilon a) \\
&= \epsilon^{n-1} \int_{S_1(a)} \int_1^\infty \mathrm{tr}_{S_1(a)} \epsilon t^{k-1} (x-a) \rfloor u (a + \epsilon t(x-a)) \wedge \phi(\epsilon(x-a) + a) dt \\
&= \epsilon^{n-1} \int_{S_1(a)} \int_\epsilon^\infty \mathrm{tr}_{S_1(a)} \epsilon^{-k+1} s^{k-1} (x-a) \rfloor u (a + s(x-a)) \wedge \phi(\epsilon(x-a) + a) ds \\
&= \epsilon^{n-k} \int_{S_1(a)} \int_\epsilon^\infty \mathrm{tr}_{S_1(a)} s^{k-1} (x-a) \rfloor u (a + s(x-a)) \wedge \phi(\epsilon(x-a) + a) ds.
\end{aligned}
$$

We make a case distinction. When $k < n$, then the double integral itself is bounded uniformly in $\epsilon > 0$ and so the last expression vanishes as $\epsilon$ goes to zero. When instead $k = n$, then the last expression equals

$$ 
\begin{aligned}
& \int_{S_1(a)} \int_{\epsilon}^{\infty} \mathrm{tr}_{S_1(a)} s^{n-1} (x-a) \rfloor u (a + s(x-a)) \wedge \phi(\epsilon(x-a) + a) ds \\
&= \int_{\mathbb{R}^n \setminus B_{\epsilon}(0)} u(y) \wedge \phi \left( \epsilon \frac{y-a}{\|y-a\|} + a \right) dy.
\end{aligned}
$$

The limit of this is $\phi(a) \int_{\Omega} u(x)$ as $\epsilon$ goes to zero. To complete the discussion, we write the exterior
derivative of $u$ as in (48), and apply the Bogovskiï potential operator to this result, which gives

$$
\begin{align*}
-B_{k+1,a} du(x) &= (x-a)\rfloor \sum_{\sigma \in \Sigma(k,n)} \sum_{j=1}^{n} \int_{1}^{\infty} t^k \frac{\partial u_{\sigma}}{\partial x_j} (a+t(x-a)) dt \, dx^j \wedge dx^{\sigma} \\
&= \sum_{\substack{\sigma \in \Sigma(k,n) \\ 1 \le j \le n, j \notin [\sigma]}} \int_{1}^{\infty} t^k \frac{\partial u_{\sigma}}{\partial x_j} (a+t(x-a)) dt \, (x-a)_j \, dx^{\sigma} \\
&\quad - \sum_{\substack{\sigma \in \Sigma(k,n) \\ 1 \le j \le n, j \notin [\sigma] \\ 1 \le i \le k}} (-1)^{i-1} \int_{1}^{\infty} t^k \frac{\partial u_{\sigma}}{\partial x_j} (a+t(x-a)) dt \, (x-a)_{\sigma(i)} \, dx^j \wedge dx^{\sigma-\sigma(i)}.
\end{align*}
$$

We add the (distributional) exterior derivative of the potential and the potential of the exterior derivative. In a manner that fully analogous to the discussion of the averaged Poincaré operator save for the modification when $k = n$, we come to the conclusion that

$$
\begin{align*}
u(x) &= dB_{k,a}u(x) + B_{k+1,a}du(x), \quad 0 \le k \le n-1, \\
u(x) &= dB_{n,a}u(x) - \left( \int_{\Omega} u(a) \right) \delta_a, \quad k = n.
\end{align*}
$$

Here, $\delta_a$ denotes the Dirac delta at $a$. In summary, after taking the average over $a \in \Omega$:

$$
u(x) = dB_k u(x) + B_{k+1} du(x), \quad 0 \le k \le n-1, \qquad (51)
$$

$$
u(x) = dB_n u(x) - \left( \text{vol}(\Omega)^{-1} \int_{\Omega} u(a) \right) \chi_{\Omega}, \quad k = n. \tag{52}
$$

6.6 Operator norms as bounds for the Poincaré–Friedrichs constants

We are now ready to state the main results of this section. Recall that $\delta(\Omega) > 0$ is the diameter of $\Omega$ and that $B_{\delta(\Omega)}(0)$ is the $n$-dimensional ball centered at the origin. The following upper bounds for the Poincaré–Friedrichs constants are proportional to the domain diameter and are independent of the Lebesgue exponent $1 \le p \le \infty$. However, the space dimension $n$ and the form degree $k$ enter the estimates, namely through definitions (46) and (47) of respectively $C_P(n, k)$ and $C_B(n, k)$.

**Theorem 6.2.** Let $\Omega \subseteq \mathbb{R}^n$ be a bounded convex open set and let $1 \le p \le \infty$. We have bounded operators

$$
\mathcal{P}_k : L^p \Lambda^k(\Omega) \to W^p \Lambda^{k-1}(\Omega), \quad \mathcal{B}_k : L^p \Lambda^k(\Omega) \to W_0^p \Lambda^{k-1}(\Omega).
$$

They satisfy the operator norm bounds

$$
\begin{align*}
\| \mathcal{P}_k u \|_{L^p(\Omega)} &\le C_{\mathrm{PF}, \mathcal{P}, \Omega, k, p} \| u \|_{L^p(\Omega)}, \\
\| \mathcal{B}_k u \|_{L^p(\Omega)} &\le C_{\mathrm{PF}, \mathcal{B}, \Omega, k, p} \| u \|_{L^p(\Omega)},
\end{align*}
$$

where

$$
C_{\mathrm{PF}, \mathcal{P}, \Omega, k, p} := C_{\mathcal{P}}(n, k) \mathrm{vol}_{n-1}(S_1(0)) \frac{\delta(\Omega)^n}{\mathrm{vol}(\Omega)} \delta(\Omega), \quad (53)
$$

$$
C_{\mathrm{PF}, \mathcal{B}, \Omega, k, p} := C_{\mathcal{B}}(n, k) \mathrm{vol}_{n-1}(S_1(0)) \frac{\delta(\Omega)^n}{\mathrm{vol}(\Omega)} \delta(\Omega). \quad (54)
$$

For any $u \in W^p \Lambda^k(\Omega)$ it holds for a.e. $x \in \Omega$ that

$$
u(x) = \mathcal{P}_1 du(x) - \text{vol}(\Omega)^{-1} \int_{\Omega} u(a) \, da, \quad k=0, \tag{55}
$$

$$
u(x) = d\mathcal{P}_k u(x) + \mathcal{P}_{k+1} du(x), \quad 1 \le k \le n. \tag{56}
$$

In particular, if $u \in W^p \Lambda^k(\Omega)$ with $0 \le k \le n-1$, then $w = \mathcal{P}_k du$ satisfies $dw = du$.
For any $u \in W_0^p \Lambda^k(\Omega)$ it holds for a.e. $x \in \Omega$ that

$$u(x) = d\mathcal{B}_k u(x) + \mathcal{B}_{k+1} du(x), \quad 0 \le k \le n-1, \qquad (57)$$

$$u(x) = d\mathcal{B}_n u(x) - \left( \text{vol}(\Omega)^{-1} \int_{\Omega} u(a) \right) \chi_{\Omega}, \quad k=n. \qquad (58)$$

In particular, if $u \in W_0^p \Lambda^k(\Omega)$ with $0 \le k \le n-1$, then $w = \mathcal{B}_k du$ satisfies $dw = du$.

**Proof.** Consider the case $1 \le p < \infty$. Because the domain is bounded, the subspaces $C_c^\infty \Lambda^k(\Omega)$ are dense in $L^p \Lambda^k(\Omega)$, and so the stated operator norm bounds follow by an approximation argument. Taking the limit on both sides of equation then implies the inequality with $p = \infty$ because the $L^\infty$ norm is the limit of the Lebesgue norms as $p$ goes to infinity.

Consider now $u \in W^p \Lambda^k(\Omega)$ with $1 \le k \le n$. We write $\zeta := \mathcal{P}_k u$. There exists a sequence $u_i \in C_c^\infty \Lambda^k(\bar{\Omega})$ that converges to $u$ in $W^p \Lambda^k(\Omega)$. For any test form $v \in C_c^\infty \Lambda^{n-k-1}(\Omega)$, we verify

$$\int_{\Omega} v \wedge u_i = \int_{\Omega} v \wedge \mathcal{P}_{k+1} du_i + \int_{\Omega} v \wedge d\mathcal{P}_k u_i \\ = \int_{\Omega} v \wedge \mathcal{P}_{k+1} du_i + (-1)^{k(n-k)+1} \int_{\Omega} dv \wedge \mathcal{P}_k u_i.$$

By the continuity of bounded linear functionals, we find

$$\int_{\Omega} v \wedge u - \int_{\Omega} v \wedge \mathcal{P}_{k+1} du = (-1)^{k(n-k)+1} \int_{\Omega} dv \wedge \mathcal{P}_k u.$$

Hence, by definition, $\zeta \in W^p \Lambda^{k-1}(\Omega)$ with $d\zeta = u - \mathcal{P}_{k+1} du$. This shows (56), and (55) follows by an approximation argument.

Analogously, suppose that $u \in W_0^p \Lambda^k(\Omega)$ with $0 \le k \le n-1$. We write $\zeta := \mathcal{B}_k u$. There exists a sequence $u_i \in C_c^\infty \Lambda^k(\bar{\Omega})$ that converges to $u$ in $W^p \Lambda^k(\Omega)$. For any test form $v \in C_c^\infty \Lambda^{n-k-1}(\Omega)$, which is the restriction of some member of $C_c^\infty \Lambda^{n-k-1}(\mathbb{R}^n)$, it holds that

$$\int_{\Omega} v \wedge u_i = \int_{\Omega} v \wedge \mathcal{B}_{k+1} du_i + \int_{\Omega} v \wedge d\mathcal{B}_k u_i \\ = \int_{\Omega} v \wedge \mathcal{B}_{k+1} du_i + (-1)^{k(n-k)+1} \int_{\Omega} dv \wedge \mathcal{B}_k u_i.$$

By the continuity of bounded linear functionals, we find

$$\int_{\Omega} v \wedge u - \int_{\Omega} v \wedge \mathcal{B}_{k+1} du = (-1)^{k(n-k)+1} \int_{\Omega} dv \wedge \mathcal{B}_k u.$$

By definition, $\zeta \in W_0^p \Lambda^{k-1}(\Omega)$ with $d\zeta = u - \mathcal{B}_{k+1} du$. This shows (57), and (58) follows by an approximation argument.

Everything else is now apparent, and the proof is complete.

□

These constants are generally not optimal. For example, when $p = 2$ and when only the divergence is considered, we have the following improved estimate.

**Lemma 6.3.** Let $\Omega \subseteq \mathbb{R}^n$ be a bounded open set. For each $u \in W^2(\text{div}, \Omega)$ there exists $w \in W^2(\text{div}, \Omega)$ with $\text{div} w = \text{div} u$ and

$$\|w\|_{L^2(\Omega)} \le \delta(\Omega) \| \text{div } u \|_{L^2(\Omega)}.$$

**Proof.** This is a reduction to the Friedrichs inequality. The space $W_0^{1,2}(\Omega)$ is the closure of the smooth functions with support in $\Omega$ in the Hilbert space $W^{1,2}(\Omega)$. Then $\nabla : W_0^{1,2}(\Omega) \subseteq L^2(\Omega) \to L^2(\Omega)$ is a closed densely-defined linear operator whose smallest singular value is bounded from below by $\delta(\Omega)^{-1}$, according to the Friedrichs inequality. The adjoint is the closed densely-defined linear operator $-\text{div} : W^2(\text{div}, \Omega) \subseteq L^2(\Omega) \to L^2(\Omega)$, which has the same smallest singular value. □
**Remark 6.4.** The classical Poincaré operator is known for its role in proving the exactness of the smooth de Rham complex over star-shaped domains [48]. The Bogovskiǐ-type operators were first studied for the divergence operator and are a staple in the mathematics of hydrodynamics [10]. Costabel and McIntosh [22] regularize the potentials by averaging over pivot points within an interior ball using a smooth compactly supported weight function, which is why they can study domains star-shaped with respect to a ball. Their operators are pseudo-differential operators of negative order, because their averaging uses a smooth weight; this proves that the operators are bounded between a variety of function spaces. Explicit bounds for the higher-order seminorms of these pseudo-differential operators have been recently contributed by Guzman and Salgado [40]. Instead, we average over the entire domain, which requires a convex geometry, and we are interested in boundedness in the Lebesgue p-norms. We establish computable bounds on the Poincaré-Friedrichs constants, which had been absent from the literature, to the best of our knowledge.

# 7 Shellable triangulations of manifolds

We return to the theory of triangulations, as our main objective requires some further concepts. We are interested in simplicial complexes that triangulate domains and that are shellable. Such shellable simplicial complexes are constructed by successively adding simplices in a well-structured manner. Local patches (stars) within triangulations of dimension two or three are examples of shellable complexes. The monographs by Kozlov [43] and Ziegler [71] are our main references for this section. We also refer to Lee's monograph [47] for any further background on manifolds.

<figure><img src="image_6.png" /><figcaption>Figure 3: Left: manifold triangulation of an annulus. Right: not a manifold triangulation.</figcaption></figure>

## 7.1 Triangulations of manifolds

Our discussion requires some notions and results concerning triangulated manifolds. We define an *n*-dimensional simplicial complex to be a *manifold triangulation* if the underlying set $|T|$ is an *n*-dimensional manifold, possibly with boundary. We recall<fnref n="5" /> that this means that for every $x \in |T|$ there exists an open neighborhood $U(x) \subseteq |T|$ and an embedding $\phi: U(x) \to \mathbb{R}^n$ such that $\phi(0) = 0$ and $\phi$ is an isomorphism either onto the open unit ball $B = \{x \in \mathbb{R}^n \mid |x| < 1\}$ or onto the half-ball $\{x \in B \mid x_1 \ge 0\}$. In the former case, *x* is called an *interior point*, and in the latter case *x* is called a *boundary point*. Any simplicial complex that triangulates an *n*-dimensional manifold must be *n*-dimensional. Figure 3 shows an example of simplicial complex that triangulates a manifold and an example that does not.

The following special cases receive particular interest: an *n*-ball triangulation is any triangulation of a topological (closed) *n*-ball, and an *n*-sphere triangulation is any triangulation of a topological *n*-sphere.

We know that any manifold $\mathcal{M}$ has got a topological boundary $\partial\mathcal{M}$, possibly empty. If $\mathcal{M}$ is *n*-dimensional, then the $\partial\mathcal{M}$ is a topological manifold without boundary of dimension *n* − 1. We gather a few helpful observations on how these notions relate to triangulations. While the reader might deem them obvious, we nevertheless include proofs.

**Lemma 7.1.** Let $\mathcal{T}$ be a finite $n$-dimensional simplicial complex whose underlying set is a manifold $\mathcal{M}$. Then the simplices contained in the boundary constitute a triangulation of the boundary. Moreover, if $F \in \mathcal{T}$ is a face (i.e, $F$ has dimension $n-1$), then

<footnote n="5">⁵In the usual terminology of topology and differential geometry, our notion of manifold is a manifold with boundary, even if the boundary is empty.</footnote>
• $\mathcal{F}$ is not contained in the boundary if and only if it is contained in exactly two $n$-simplices.

• $\mathcal{F}$ is contained in the boundary if and only if it is contained in exactly one $n$-simplex.

*Proof.* We prove these statements in several steps.

1. Let $\mathcal{M} := \mathcal{M} \setminus \partial\mathcal{M}$ denote the interior of the manifold. We will use the following fact:<fnref n="6" /> if $S \in \mathcal{T}$ has an inner point that lies on $\partial\mathcal{M}$, then all inner points of $S$ are on $\partial\mathcal{M}$. Since the boundary $\partial\mathcal{M}$ is closed, every $S \in \mathcal{T}$ is either a subset of the boundary or all its inner points lie in the interior $\mathcal{M}$ of the manifold.

2. We recall an auxiliary result. Suppose that $Y$ is a topological space homeomorphic to a sphere of dimension $m$ and that $X \subseteq Y$ is homeomorphic to a sphere of dimension $m-1$, where $m \ge 1$. As a consequence of the Jordan–Brouwer separation theorem [56, Corollary IV.5.24], [54, Corollary VIII.6.4], we know that $Y \setminus X$ has got two connected components.

3. Let now $F \in S_{n-1}^{\perp}(T)$ be a face and let $z_F \in F$ be its barycenter. Since $\mathcal{T}$ is finite, we let $\mathring{B}_F$ be an open neighborhood around $z_F$ homeomorphic to an $n$-dimensional ball so small that its closure $\bar{B}_F$ only intersects those $n$-simplices of $\mathcal{T}$ that already contain $z_F$ and no faces other than $F$. Suppose there are distinct $n$-simplices $T_1, T_2, \dots, T_K$ that contain $z_F$. The intersection of any two of them is $F$, but their interiors are disjoint because otherwise they would coincide.

If $z_F$ is an interior point of $\mathcal{M}$, then it follows by our assumptions that $\mathring{B}_F$ is homeomorphic to an open $n$-ball and $\partial\mathring{B}_F$ is homeomorphic to a sphere of dimension $n-1$. Consider $X = F \cap \partial\mathring{B}_F$. If $n=1$, then $X$ is empty and $\partial\mathring{B}_F$ has $K$ distinct connected components. If $n > 1$, then $X$ is homeomorphic to a sphere of dimension $n-2$ and again $\partial\mathring{B}_F \setminus X$ has $K$ distinct connected components. But by the auxiliary result above, $K=2$. We conclude that $F$ is contained in two $n$-simplices of $\mathcal{T}$.

Consider the case that $z_F$ lies on the boundary of $\mathcal{M}$ and suppose that $F$ is contained in $K$ distinct $n$-simplices of $\mathcal{T}$. By adding at least one dimension, we can double<fnref n="7" /> the manifold $\mathcal{M}$ along the boundary and obtain the doubled manifold $\mathcal{M}'$. Similarly, we can construct a doubling of the triangulation $\mathcal{T}'$ such that $F$ is contained in exactly $2K$ distinct $n$-simplices of $\mathcal{T}'$. We know that $\mathcal{M}'$ is a manifold without boundary, and hence $F$ is an interior simplex of $\mathcal{T}'$. This implies $K=1$. So any boundary face can only be contained in one single $n$-simplex.

4. Clearly, the simplices of $\mathcal{T}$ contained in the boundary constitute a simplicial complex. Every $x \in \mathcal{M}$ is an inner point of some simplex $S \in \mathcal{T}$. If $x \in \partial\mathcal{M}$ is a boundary point, then $S$ must be a boundary simplex, so the boundary simplices triangulate all of $\partial\mathcal{M}$.

All desired results are thus proven. $\square$

<figure><img src="image_7.png" /></figure>

Suppose that $\mathcal{T}$ is an $n$-dimensional simplicial complex that triangulates a manifold. Those simplices of the manifold triangulation that are subsets of the boundary of the underlying manifold are called *boundary simplices*. All other simplices of the manifold triangulation are called *inner simplices*. We have seen that the boundary simplices of a manifold triangulation of the manifold's boundary. We call this simplicial complex the *boundary complex*. It has dimension $n-1$.

We continue with a few more observations about the topology of local patches (stars) of manifold triangulations. This topic is surprisingly non-trivial. We only gather some results that are hard to find in the literature.

**Lemma 7.2.** Let $\mathcal{T}$ be a finite $n$-dimensional simplicial complex whose underlying space is a manifold $\mathcal{M}$. Suppose that $1 \le n \le 3$. Then the following holds:

• If $S \in \mathcal{T}$ is an inner simplex, then $\mathrm{st}_{\mathcal{T}}(S)$ is a simplicial $n$-ball with $S$ as an inner simplex and $\partial\mathrm{st}_{\mathcal{T}}(S)$ is a simplicial $(n-1)$-sphere.

<footnote n="6">⁶To see this, one easily constructs a continuous deformation of $\mathcal{M}$ into itself to move any chosen point on $S$ to any other chosen point on $S$.</footnote>

<footnote n="7">⁷The reader is referred to Lee's textbook [48] for more background and the technicalities.</footnote>
* If $S \in \mathcal{T}$ is a boundary subsimplex, then $\mathrm{st}_{\mathcal{T}}(S)$ is a simplicial $n$-ball with $S$ as a boundary simplex, and $\partial \mathrm{st}_{\mathcal{T}}(S)$ is a simplicial $(n-1)$-ball.

*Proof.* The statement is obvious if $n=1$, so we assume $n \ge 2$ in what follows. We prove these statements in several steps. The reader is assumed to have some background in topology.

1. Let $S$ be any simplex with vertices $v_0, v_1, \dots, v_k$, with barycenter $z_S$, and dimension $k$. Let $S := \mathrm{st}_{\mathcal{T}}(S)$ be its star. Each $l$-dimensional simplex $T \in S$ that contains $S$ has vertices $v_0, v_1, \dots, v_k, v_{k+1}^S, \dots, v_l^S$. For any such simplex, we introduce a decomposition $T_0, \dots, T_k$, where each $T_i$ has vertices $v_0, \dots, v_{i-1}, z_S, v_{i+1}, \dots, v_k, v_{k+1}^S, \dots, v_l^S$. The collection $S^*$ of these simplices and their subsimplices constitute a simplicial complex that triangulates the same underlying set as $S$. Moreover, $S^* = \mathrm{st}_{S^*}(z_S)$. In particular, $z_S$ is a boundary vertex of $S^*$ if and only if $S$ is a boundary simplex of $S$. So it remains to study the topology of vertex stars.

2. Suppose that $2 \le n \le 3$ and that $\mathcal{M}$ is a manifold without boundary. Under these assumptions, as explained in the proof of Theorem 1 in [63], the set $\partial \mathrm{st}_{\mathcal{T}}(V)$ is a triangulation of a sphere of dimension $n-1$ for any inner vertex $V$. There exists a homeomorphism from the closed cone of $|\partial \mathrm{st}_{\mathcal{T}}(V)|$ onto the local star $|\mathrm{st}_{\mathcal{T}}(V)|$. But then that closed cone and hence $|\mathrm{st}_{\mathcal{T}}(V)|$ are homeomorphic to an $n$-dimensional ball.

3. If $2 \le n \le 3$ and $\mathcal{M}$ has a non-empty boundary, then we use an approach as in the proof of Lemma 7.1: we let $\mathcal{M}'$ denote the doubling of the manifold and $\mathcal{T}'$ be the doubling of the triangulation $\mathcal{T}$. Let $V \in \mathcal{T}$ be a vertex. If $V$ is an inner vertex of $\mathcal{T}$, then $\partial \mathrm{st}_{\mathcal{T}}(V) \subseteq \mathcal{T} \subseteq \mathcal{T}'$ triangulates a sphere of dimension $n-1$ and $\mathrm{st}_{\mathcal{T}}(V) \subseteq \mathcal{T} \subseteq \mathcal{T}'$ triangulates a ball of dimension $n$, as discussed above. If $V$ is a boundary vertex of $\mathcal{T}$, then it is an inner vertex of $\mathcal{T}'$, and so $\partial \mathrm{st}_{\mathcal{T}}(V) \subseteq \mathcal{T}'$ triangulates a sphere of dimension $n-1$ and $\mathrm{st}_{\mathcal{T}}(V) \subseteq \mathcal{T}'$ triangulates a ball of dimension $n$. We also know that $\partial \mathrm{st}_{\partial \mathcal{T}}(V) \subseteq \partial \mathcal{T}$ triangulates a sphere of dimension $n-2$ and $\mathrm{st}_{\partial \mathcal{T}}(V) \subseteq \partial \mathcal{T}$ triangulates a ball of dimension $n-1$. The embedding of $\partial \mathrm{st}_{\partial \mathcal{T}}(V) \subseteq \partial \mathcal{T}$ is homeomorphic to the standard embedding of the $(n-2)$-dimensional unit sphere into the $(n-1)$-dimensional unit sphere, by the topological Schoenflies theorem [58] It follows that $\partial \mathrm{st}_{\mathcal{T}}(V)$ triangulates a topological ball of dimension $n-1$. Since the closed cone of $|\partial \mathrm{st}_{\mathcal{T}}(V)|$ is homeomorphic to the star $|\mathrm{st}_{\mathcal{T}}(V)|$, we conclude that $\mathrm{st}_{\mathcal{T}}(V)$ triangulates an $n$-dimensional ball.

All relevant results are proven. $\square$

**Lemma 7.3.** Let $\mathcal{T}$ be a finite $n$-dimensional simplicial complex whose underlying space is a manifold $\mathcal{M}$. If the underlying space of $\mathcal{T}$ is connected, then $\mathcal{T}$ is face-connected.

*Proof.* We first show that each vertex star is face-connected via a short induction argument. Clearly, any simplicial 1-ball and simplicial 1-sphere are face-connected. Now, if $n \ge 1$ and $V \in \mathcal{T}$, then the $n$-simplices in $\mathrm{st}_{\mathcal{T}}(V)$ are in correspondence to the $(n-1)$-simplices in $\partial \mathrm{st}_{\mathcal{T}}(V)$. Hence, $\mathrm{st}_{\mathcal{T}}(V)$, a triangulation of dimension $n$, is face-connected if and only if $\partial \mathrm{st}_{\mathcal{T}}(V)$, a triangulation of dimension $n-1$, is face-connected. The induction argument implies that each vertex star in $\mathcal{T}$ is face-connected.

We assume that the underlying space $|\mathcal{T}|$ is connected, and hence path-connected. Given $n$-simplices $S, S' \in \mathcal{T}$, there exists a path $\gamma : [0, 1] \to |\mathcal{T}|$ from the barycenter of $S$ to the barycenter of $S'$. We can choose a sequence of $n$-simplices $S = \hat{S}_0, \hat{S}_1, \dots, \hat{S}_m = S' \in \mathcal{T}$ without repetitions that cover the path and whose intersections with the path are homeomorphic to $[0, 1]$. For any index $1 \le i \le m$, the intersection $\gamma([0, 1]) \cap S_i$ and $\gamma([0, 1]) \cap S_{i+1}$ intersect at one point lying in a common subsimplex of $S_i$ and $S_{i+1}$. We deduce that each two consecutive simplices in the sequence $\hat{S}_0, \hat{S}_1, \dots, \hat{S}_m$ will have at least one vertex in common. As each vertex star is face-connected, there exists a sequence $S = S_0, S_1, \dots, S_m = S' \in \mathcal{T}$ where $S_i \cap S_{i-1}$ is a face of both $S_i$ and $S_{i-1}$ for all $1 \le i \le m$. This just means that $\mathcal{T}$ is face-connected. $\square$

**Remark 7.4.** Triangulations with the property that all vertex stars are homeomorphic to a ball are also called combinatorial [8, Section 1]. All manifolds of dimension up to three admit smooth structures and smooth manifolds admit combinatorial triangulations. There are triangulations of manifolds in more than three dimensions where not every vertex star is homeomorphic to a ball.

Not every simplicial complex is the triangulation of some (embedded) topological manifold with or without boundary. When the dimension is at least five, then there are manifolds for which no computer
algorithm, given a finite simplicial complex as input, can decide whether the input is the triangulation of that manifold [20]. Going further, it has been shown that deciding whether a simplicial complex triangulates any manifold cannot be decided by any computer algorithm [61]. We therefore are not in pursuit of any easy combinatorial property that indicates whether a simplicial complex (without any further specific assumptions) triangulates a manifold.

Conversely, not all topological manifolds, even if compact, can be described as a triangulation. Such manifolds, some even compact and simply-connected, appear in dimension four and higher [3].

## 7.2 Shellable simplicial complexes

The notions of shelling and shellable triangulation have been discussed widely in combinatorial topology and polytopal theory. Formally, a triangulation is shellable if its full-dimensional simplices can be enumerated such that each simplex intersects the union of the previously listed simplices in a codimension one triangulation of a manifold. This forces the intermediate triangulations to be particularly well-shaped. We build upon the notion of shelling as introduced in [71, Definition 8.1], where our definition of shelling is equivalent to the notion of the shellings of simplicial complexes, see also [71, Remark 8.3].

Suppose that $\mathcal{T}$ is an $n$-dimensional simplicial complex and that we have an enumeration of the $n$-simplices $T_0, T_1, T_2, \dots \in S_n^*(\mathcal{T})$. For any enumeration, we call

$$ \Gamma_m := (T_0 \cup T_1 \cup \dots \cup T_{m-1}) \cap T_m, \quad 1 \le m, $$

the $m$-th interface set. We call the enumeration a shelling if each interface set $\Gamma_m$ is a triangulated manifold of dimension $n-1$.

**Remark 7.5.** In our setting, this is equivalent to the interface set $\Gamma_m$ being a non-empty collection of faces of the simplex $T_m$. In particular, for $n=2$, this interface $\Gamma_m$ is a collection of edges and cannot include single vertices of $T_m$, and for $n=3$, the interface $\Gamma_m$ is a collection of faces that cannot include any single vertices or single edges of $T_m$.

**Remark 7.6.** We interpret a shelling as the construction of a triangulation by successively attaching simplices such that the intermediate triangulations are well-behaved. Conversely, the reverse enumeration describes a successive decomposition of the triangulation, hence the name “shelling”.

We now study some major consequences of the definition of shellable simplicial complexes. Shellable simplicial complexes are constructed via successive adhesion of simplices. The resulting succession of intermediate simplicial complexes consists of simplicial balls or spheres. In particular, this applies to the full triangulation. Consequently, a domain with a shellable triangulation must be contractible.

**Lemma 7.7.** Let $\mathcal{T}$ be an $n$-dimensional simplicial complex with a shelling $T_0, T_1, T_2, \dots, T_M$, such that each simplex of dimension $n-1$ is contained in at most two simplices. Then

$$ X_m := T_0 \cup T_1 \cup \dots \cup T_m $$

is a triangulated manifold, possibly with boundary, for all $0 \le m \le M$. In particular, $X_m$ is a topological $n$-ball when $m < M$, and $X_M$ is either a topological $n$-ball or topological $n$-sphere.

*Proof.* We prove this claim by induction. Certainly, if $\mathcal{T}$ contains only one single $n$-simplex, then it is a shellable triangulation of a topological $n$-ball. Next, let $1 \le m \le M$ and suppose that

$$ X_{m-1} := T_0 \cup T_1 \cup \dots \cup T_{m-1} $$

is already known to be a topological $n$-ball. Let $T_m$ be the next $n$-simplex in the shelling. By definition, $\Gamma_m := X_{m-1} \cap T_m$ is a submanifold of $\partial T_m$, triangulated by some faces of $T_m$ and their subsimplices.

Let $F$ be such a face. By assumption, $F$ must be contained in exactly one $n$-simplex of the partial sequence $T_0, T_1, \dots, T_{m-1}$, and $F$ is in the boundary of $X_{m-1}$. We conclude that $\Gamma_m$ triangulates a submanifold of the boundary of $X_{m-1}$.

On the one hand, if $\Gamma_m$ is the entire boundary of $T_m$, then it is a topological sphere of dimension $n-1$. Since $\Gamma_m$ is a submanifold of the boundary of $X_{m-1}$, which by induction assumption is also a topological sphere of dimension $n-1$, we conclude that $\Gamma_m$ is the whole boundary of $X_m$. By basic geometric topology, $X_m$ is a topological $n$-sphere, and $T_m$ can only be the last simplex in the shelling, $m=M$. On the other hand, if $\Gamma_m$ is a proper subset of the entire boundary of $T_m$, then $X_{m-1} \cup T_m$ is still a topological $n$-ball. $\square$
Each time an $n$-simplex is added, a local star in the full triangulation around some lower-dimensional simplex is completed.

**Lemma 7.8.** Suppose that an $n$-dimensional manifold triangulation $\mathcal{T}$ has a shelling $T_0, T_1, T_2, \dots, T_M$. For $0 \le m \le M$, write

$$X_m := T_0 \cup T_1 \cup \dots \cup T_m.$$

For $1 \le m \le M$, write

$$\Gamma_m := X_{m-1} \cap T_m.$$

Then $\Gamma_m$ is a union of $\ell$ different faces of $T_m$, $1 \le \ell \le n+1$. If $m < M$, then the intersection of those faces is an interior simplex $S_m \in \mathcal{T}$ of dimension $n - \ell$ that satisfies

$$\mathrm{st}_{X_m}(S_m) = \mathrm{st}_{\mathcal{T}}(S_m).$$

*Proof.* By definition, $\Gamma_m$ is a triangulated submanifold of the boundary of $T_m$, and so it must be a collection of $\ell$ faces of $T$, $1 \le \ell \le n+1$. Note that $\ell = n+1$ can only happen for the last enumerated simplex, $m = M$, if $\mathcal{T}$ triangulates an $n$-sphere. $\Gamma_m$ also constitutes a local patch (star) of $(n-1)$-dimensional simplices around some simplex $S_m$ of dimension $n-\ell$ in $\Gamma_m$. By definition, $S_m$ is a boundary simplex of $X_m$, and it is an interior simplex of $X_{m+1}$. But then $S_m$ cannot be a subsimplex of any of the simplices $T_{m+1}, \dots, T_M$, which means that $\mathrm{st}_{X_{m+1}}(S_m) = \mathrm{st}_{\mathcal{T}}(S_m)$. $\square$

We collect important examples and counterexamples of shellable triangulations. Essentially, in two space dimensions, interesting triangulations are shellable, but non-shellable triangulations occur in dimension three and higher. In practice, triangulations of domains in 3D are shellable except for special cases that we consider as rare. Our main interest are local patches (stars) within triangulations: these are shellable up to three space dimensions.

**Example 7.9.** Any simplex $T$ (trivially) has a shelling, consisting only of $T$ itself. The boundary complex $\partial\mathcal{T}(T)$ has a shelling: any enumeration of the boundary faces of $T$ constitutes a shelling; see Example 8.2.(iii) in [71].

**Example 7.10.** The standard triangulation of the 3-dimensional cube by six tetrahedra, the Kuhn triangulation [44], is shellable, as are its higher-dimensional generalizations.<fnref n="8" />

**Example 7.11.** There exists a non-shellable triangulation of a tetrahedron and of a cube in $n = 3$, see [71, Example 8.9].

**Lemma 7.12.** Any simplicial 2-ball is shellable. Any simplicial 2-sphere is shellable.

*Proof.* First, let $S$ be any triangulation of a 2-sphere. By removing any triangle $S \in S$, we obtain a triangulation $\mathcal{T}$ of a 2-ball. Any shelling of that triangulation $\mathcal{T}$ can be extended to a shelling of $S$ by re-inserting the first triangle $S$. So it remains to show that any triangulation $\mathcal{T}$ of a two-dimensional ball is shellable. We will construct the shelling in reverse.

Write $M = |\mathcal{T}|$. There is nothing to show if $\mathcal{T}$ contains only one triangle. We call a triangle $T \in \mathcal{T}$ good in $\mathcal{T}$ if it intersects the boundary $\partial M$ in a non-empty union of edges. Hence, a triangle is good in $\mathcal{T}$ if its intersection with $\partial M$ is either one, two, or three edges, and a triangle is not good in $\mathcal{T}$ if that intersection is either empty, only some of its vertices, or a vertex and the opposite edge. We show by an induction argument over the number of triangles that every triangulation of a 2-ball that contains at least two triangles also contains at least two good triangles.

Clearly, this is the case if the triangulation of the 2-ball contains two triangles. Now suppose the induction claim is true when the triangulation includes at most $N$ triangles, and assume that $\mathcal{T}$ includes $N+1$ triangles. As we travel along the boundary, we traverse along edges of at least two simplices, and therefore there are at least two triangles with an edge on the boundary. Suppose that $\mathcal{T}$ does not have at least two triangles that are good in $\mathcal{T}$. Then there exists a triangle $T'$ that intersects $\partial M$ in one edge and its opposite vertex. Removing $T'$ splits the manifold into two face-connected components, each of which is a topological 2-ball. By the induction assumption, each of those components contains at least two triangles that are good in the respective component. So each component has at least one triangle that is also good in $\mathcal{T}$. Hence, $\mathcal{T}$ contains two good triangles, which completes the induction step.

<footnote n="8">⁸We remark that Kuhn attributes this triangulation to Lefschetz [49].</footnote>
We conclude that whenever $\mathcal{T}$ triangulates a 2-ball, it contains a good triangle $T$. If $T$ has three edges in the boundary, then $T = M$ and we are trivially done. If $T$ intersects with the boundary in exactly one or two edges, then $M \setminus T$ is still a topological 2-ball. The triangulation $\mathcal{T}'$ that is obtained by removing $T$ is a triangulation of some 2-ball that intersects $T$ only at either two or one edges. Any shelling of $\mathcal{T}'$ can in this way be extended to a shelling of $\mathcal{T}$, and the proof is complete. $\square$

**Lemma 7.13.** Let $\mathcal{T}$ be a 3-dimensional manifold triangulation and $S \in \mathcal{T}$. Then $\text{st}_{\mathcal{T}}(S)$ is shellable.

*Proof.* The statement is trivially true if $S$ is a tetrahedron. The statement is clear if $S$ is an inner or boundary face of $\mathcal{T}$, where we only need to enumerate either one or two tetrahedra. The statement is still easily verified if $S$ is an inner or boundary edge of $\mathcal{T}$: one chooses a starting tetrahedron (with a boundary face if $S$ is a boundary edge) and rotates around the edge in a fixed direction to create a suitable enumeration. When $S$ is an inner vertex, then the faces (triangles) of $\text{st}_{\mathcal{T}}(S)$ that do not contain $V$ constitute a simplicial 2-sphere. Similarly, when $S$ is a boundary vertex, then the faces (triangles) of $\text{st}_{\mathcal{T}}(S)$ that do not contain $V$ constitute a simplicial 2-ball. Both these 2-dimensional complexes are shellable by Lemma 7.12, and any such shelling immediately yields a shelling of $\text{st}_{\mathcal{T}}(S)$ since there is a one-to-one correspondence between the tetrahedra in $\mathcal{T}$ and the triangles.<fnref n="9" /> $\square$

**Lemma 7.14.** Let $\mathcal{T}$ be an $n$-dimensional shellable triangulation and $V \in \mathcal{V}(\mathcal{T})$ be a vertex. Then $\text{st}_{\mathcal{T}}(V)$ is shellable.

*Proof.* This is Lemma 8.7 in [71]. $\square$

**Lemma 7.15.** The barycentric refinement of any triangulation of a three-dimensional convex domain is shellable. The second barycentric refinement of any triangulation of a convex domain is shellable.

*Proof.* See [2, Theorem A]; see also [14, Proposition 1] $\square$

**Lemma 7.16.** Let $\mathcal{M}$ be a 3-dimensional manifold homeomorphic to a 3-simplex. Then any triangulation of $\mathcal{M}$ has a shellable refinement.

*Proof.* Let $\mathcal{T}$ be a triangulation of $\mathcal{M}$ and let $\mathcal{T}_0$ be a triangulation of the 3-simplex. According to the three-dimensional “Hauptvermutung” [57, Theorem 4], the triangulations $\mathcal{T}$ and $\mathcal{S}$ admit respective refinements $\mathcal{T}'$ and $\mathcal{S}'$ such that $\mathcal{T}$ is isomorphic to $\mathcal{S}$. The first barycentric refinement of $\mathcal{S}'$ is shellable [2, Theorem A]. We conclude that $\mathcal{T}$ has a refinement that is shellable. $\square$

**Remark 7.17.** Not all triangulable sets admit a triangulation that is also shellable. Even if a set admits some shellable triangulation, some of its other triangulations might not be shellable. We refer to [71, Example 8.9] for examples of non-shellable triangulations of cubes and tetrahedra in three dimensions. Moreover, if we extend the non-shellable triangulation of the tetrahedron from [71, Example 8.9] to a triangulation of a hypertetrahedron by suspending it from a new point $v^*$, then the resulting new triangulation is non-shellable and coincides with the patch around $v^*$. This demonstrates that patches around boundary simplices are not necessarily shellable when the space dimension $n$ is larger than three.

**Remark 7.18.** Whether a simplicial complex is shellable can be checked, in principle, simply by trying out all the possible enumerations. That we cannot do much better than this is captured in the fact that testing general simplicial complexes (with some fixed dimension) for shellability is NP-complete [35]. This complexity class even holds if we only test two-dimensional simplicial complexes for shellability, or if we only test shellability of contractible three-dimensional simplicial complexes.

# 8 Reflections and deformations on shellable stars

This section is devoted to geometric operations that are crucial for our main result in Section 9 below. Consider the situation where we have an $n$-dimensional local patch (star) around some simplex $S$ and some $n$-dimensional simplex $T$ within that local star. We construct a homeomorphism going from the simplex $T$ onto its complement within the local star around $S$. This homeomorphism, which we interpret as a nonlinear reflection, preserves the interface. We ensure that the homeomorphism is bi-Lipschitz,

<footnote n="9">⁹For $S$ an inner vertex, [25, Lemma B.1] also yields the claim.</footnote>
Figure 4: Illustration of Lemma 8.1. Left: the triangle $T = [v_0, v_1, v_2]$ is bisected at the edge $S = [v_0, v_1]$, leading to two new triangles. The height vector to $v_2$ in all three triangles remains the same. The height vector to $z_S$ in the new triangle $[z_S, v_1, v_2]$ is one half of the height vector to $v_0$ in the original triangle. Right: the triangle $T$ is trisected, leading to three new triangles. The height vector to $z_T$ from the opposite edge in any triangle is one third of the original height vector of that edge.

<figure><img src="image_8.png" /><figcaption>Figure 5: Illustration of Proposition 8.2 and its proof: for a convex vertex star, for a non-convex vertex star, both cases where the barycentric refinement of $S$ has no effect, for a convex edge star, and for a non-convex edge star.</figcaption></figure>

<figure><img src="image_9.png" /></figure>

and we are particularly interested in the norms of its Jacobian. This nonlinear reflection, which is similar to the two- and three-color maps in [25, Sections 5.3 and 6.3] and the symmetrization maps in [19, Section 7.6], will be used subsequently in generalizing the discussion in Section 4 to the setting of differential forms.

The following observation, which we state without proof, controls the volume and some heights when a simplex is partitioned via barycentric subdivision.

**Lemma 8.1.** Let $T$ be an $n$-dimensional simplex with an $\ell$-dimensional subsimplex $S$ ($\ell \ge 0$) and let $z_S$ be the barycenter of $S$. Let $T'$ be one of the $n$-dimensional simplices obtained by splitting $T$ in accordance with the barycentric subdivision of $S$ at $z_S$.

* $\text{vol}(T') = \text{vol}(T)/(\ell + 1)$.

* The height vector of $v \in \mathcal{V}(T) \setminus \mathcal{V}(S)$ in $T'$ is the height vector of $v$ in $T$.

* The height vector of $z_S$ in $T'$ is the height vector of the single vertex $v \in \mathcal{V}(T) \setminus \mathcal{V}(T')$ in $T$, scaled by $(\ell + 1)^{-1}$. $\square$

The following auxiliary result on the geometry of local stars is of independent interest.

**Proposition 8.2.** Let $\mathcal{T}$ be a triangulation of an $n$-dimensional domain. Let $S \in \mathcal{T}$ be an inner simplex of dimension $\ell < n$. Then $|\text{st}_{\mathcal{T}}(S)|$ is star-shaped with respect to the ball $B_{h/(\ell+1)}(z_S)$, where $z_S$ denotes the barycenter of $S$, and where $h > 0$ denotes the maximum height of any vertex of $S$ within any $n$-simplex of $\text{st}_{\mathcal{T}}(S)$.

*Proof.* Let $\mathcal{R}$ denote the simplicial complex obtained from $\text{st}_{\mathcal{T}}(S)$ by barycentric refinement of $S$. Then $\mathcal{R}$ is its own vertex star around $z_S$. Taking into account Lemma 8.1, the desired result follows for $\ell > 0$ if we show it for $\ell = 0$.
<figure><img src="image_10.png" /><figcaption>Figure 6: Sketch of the geometric situation in the proof of Proposition 8.3. The simplex T completes the star around an interior vertex, and U is the complement of the simplex T within that star. Here, $\Gamma_1$ denotes the interface between T and U. The reflection $\Xi_1$ maps T into U while being the identity the interface $\Gamma_1$. The mapping $\Xi_1$ is the identity outside of $\mathcal{K}$ and $\mathcal{K}^c$, only affecting the latter two sets.</figcaption></figure>

So we consider the case $\ell = 0$. Any codimension one boundary face $F \in \mathrm{st}_{\mathcal{T}}(S)$ corresponds to some $n$-simplex $T \in \mathrm{st}_{\mathcal{T}}(S)$, where $F$ is the face of $T$ opposite to its vertex $z_S$. For any such codimension one boundary face $F \in \mathrm{st}_{\mathcal{T}}(S)$, we let $H_F$ denote its affine hull, which is a half-plane.

Assume now that $\varrho > 0$ is at most the minimum height of $z_S$ in any $n$-simplex $T \in \mathrm{st}_{\mathcal{T}}(S)$. If $x \in \mathcal{B}_{\varrho}(z_S)$ and $F \in \mathrm{st}_{\mathcal{T}}(S)$ is a boundary face, then our choice of $\varrho$ implies that $x$ and $z_S$ must lie on the same side of $H_F$. In particular, $\mathcal{B}_{\varrho}(z_S)$ cannot intersect any boundary face of $\mathrm{st}_{\mathcal{T}}(S)$, which shows $\mathcal{B}_{\varrho}(z_S) \subseteq |\mathrm{st}_{\mathcal{T}}(S)|$.

To show that $|\mathrm{st}_{\mathcal{T}}(S)|$ is star-shaped with respect to $\mathcal{B}_{\varrho}(z_S)$, it suffices to show that for every two-dimensional hyperplane $P$ through $z_S$ the intersection $|\mathrm{st}_{\mathcal{T}}(S)| \cap P$ is star-shaped with respect to $\mathcal{B}_{\varrho}(z_S) \cap P$. The latter is deduced observing that $|\mathrm{st}_{\mathcal{T}}(S)| \cap P$ is a two-dimensional polygon within the plane $P$ and using a well-known result of planar geometry [46]. $\square$

We now provide the desired bi-Lipschitz transformation, which maps a local star onto itself. We interpret it as a nonlinear reflection across the interface between some fixed simplex of the local star and the remainder of that star. We give detailed estimates for the singular values of the Jacobian.

**Proposition 8.3.** Let $\mathcal{T}$ be a triangulation of an $n$-dimensional domain. Let $S \in \mathcal{T}$ be an inner simplex of dimension $\ell < n$, let $T \in \mathrm{st}_{\mathcal{T}}(S)$ be of dimension $n$, and let

$$U := |\mathrm{st}_{\mathcal{T}}(S)| \setminus T, \quad \Gamma_1 := U \cap T.$$

The following holds, where the constants on the right-hand sides are as stated in the proof. There exists a bi-Lipschitz piecewise affine mapping

$$\Xi_1 : T \to \Xi_1(T) \subseteq U$$

which is the identity along $\Gamma_1 = T \cap U$. At any point, the singular values $\sigma_1 \ge \dots \ge \sigma_n$ of its Jacobian satisfy

$$\sigma_1 \le C_{5,n,\ell}(T), \quad \sigma_2, \dots, \sigma_n \le C'_{5,n,\ell}(T), \quad |\det \mathbf{J}\Xi_1| \le C^{\mathrm{det}}_{5,n,\ell}(T), \\ \sigma_n^{-1} \le C_{6,n,\ell}(T), \quad \sigma_1^{-1}, \dots, \sigma_{n-1}^{-1} \le C'_{6,n,\ell}(T), \quad |\det \mathbf{J}\Xi_1^{-1}| \le C^{\mathrm{det}}_{6,n,\ell}(T).$$

Moreover, for any $0 \le k \le n$ and $p \in [1, \infty]$,

$$\sigma_1 \cdots \sigma_k |\det \mathbf{J}\Xi_1|^{-\frac{1}{p}} \le C_{5,n,k,\ell,p}(T), \quad \sigma_n^{-1} \cdots \sigma_{n-k+1}^{-1} |\det \mathbf{J}\Xi_1|^{\frac{1}{p}} \le C_{6,n,k,\ell,p}(T).$$

*Proof.* We derive the estimate in several steps. In what follows, we use the notation $\hat{z}$ for the normalization of any vector $z \in \mathbb{R}^n$.

* Without loss of generality, after shifting the coordinate system, the barycenter $z_S$ of $S$ is the origin, $z_S := 0$.

We fix the subsimplex $S' \subseteq T$ that is complementary to $S$ within the simplex $T$. Note that $\Gamma_1 = U \cap T$ is the union of exactly those faces of $T$ that contain $S$, We let $z_{S'}$ be the midpoint of $S'$.

We apply barycentric refinement to the simplex $S$ and then the complementary simplex $S'$. This produces a new triangulation $\mathcal{K}$ of $T$ whose $n$-dimensional simplices contain the vertices $z_S$ and $z_{S'}$.
* We use Proposition 8.2: st<sub>T</sub>(S) is star-shaped with respect to the closed ball $\overline{\mathcal{B}}_\rho(z_S)$, where $\rho = h/(\ell+1)$ and h is the maximum height of any vertex of S within any n-simplex $T \in \text{st}_T(S)$.

* Let $\rho \in (0, 1]$, yet to be determined. We define $y := -\rho z_{S'}$ as vector in the opposite direction of $z_{S'}$ and with length $\rho\|z_{S'}\|$. Henceforth, we assume $\rho \le \rho/\|z_{S'}\|$ so that $y \in \overline{\mathcal{B}}_\rho(z_S)$. By construction, the closed line segment from $z_S$ to $y$ is contained in $\overline{\mathcal{B}}_\rho(z_S)$, and we conclude that the convex closure of $T$ and $y$ must lie in st<sub>T</sub>(S).

We define another triangulation $\mathcal{K}^c$ as follows: given any $n$-simplex $K \in \mathcal{K}$, we replace its vertex $z_S'$ by the vertex $y$ at the opposite position, thus obtaining a new $n$-simplex $K^c$. Indeed, that $\mathcal{K}^c$ is a simplicial complex follows easily from $\mathcal{K}$ being a simplicial complex.

We let the simplicial complex $\mathcal{K}^*$ be the union of $\mathcal{K}$ and $\mathcal{K}^c$. By construction, all its $n$-simplices contain $z_S$ as a subsimplex, which is an inner vertex of that triangulation. In particular, $\mathcal{K}^*$ is its own star around $z_S$.

* We introduce a new mapping $\Theta: |\mathcal{K}| \to |\mathcal{K}^c|$ between the underlying sets as follows: if $K \in \mathcal{K}$ be an $n$-simplex and $K^c \in \mathcal{K}^c$ is constructed from $K$, then $\Theta|_K$ is the unique affine bijection that maps the vertex $z_{S'}$ to the vertex $y$ and leaves all other vertices the same. It follows that

$$|\det(\mathbf{J}\Theta|_K)| = \frac{\text{vol}(K^c)}{\text{vol}(K)}.$$

We want to further characterize the singular values of this transform's Jacobian. Let $h_z \in \mathbb{R}^n$ be the height vector of $z_{S'}$ inside the simplex $K$, that is, the vector pointing to $z_{S'}$ and standing orthogonally on the affine hull of the face $K$ that opposes $z_{S'}$. Let us verify that

$$\Theta|_K(x) = x - (1 + \rho) \frac{\langle \hat{h}_z, x \rangle}{\langle \hat{h}_z, \hat{z}_{S'} \rangle} \hat{z}_{S'}.$$

Indeed, the right-hand side equals $x$ whenever $x$ lies in the plane orthogonal to $h_z$, and when $x = z_{S'}$, then

$$\begin{aligned} \Theta|_K(z_{S'}) &= z_{S'} - (1 + \rho)\hat{z}_{S'} \frac{\langle \hat{h}_z, z_{S'} \rangle}{\langle \hat{h}_z, \hat{z}_{S'} \rangle} \\ &= z_{S'} - (1 + \rho)\hat{z}_{S'} \|z_{S'}\| = z_{S'} - (1 + \rho)z_{S'} = -\rho z_{S'} = y. \end{aligned}$$

If we orthogonally decompose $z_{S'} = h_z + b_z$, where $b_z \in \mathbb{R}^n$, then

$$\begin{aligned} \Theta|_K(h_z) &= h_z - (1 + \rho) \frac{\langle h_z, h_z \rangle}{\langle h_z, z_{S'} \rangle} z_{S'} \\ &= h_z - (1 + \rho) \frac{\langle h_z, z_{S'} \rangle}{\langle h_z, z_{S'} \rangle} z_{S'} = h_z - (1 + \rho)z_{S'} = -\rho h_z - (1 + \rho)b_z. \end{aligned}$$

Evidently, the transformation $\Theta|_K$ equals the identity on the orthogonal complement of the span of $h_z$ and $b_z$. Let $\beta$ be the angle between $z_{S'}$ and $h_z$. Then $\|h_z\| = \cos(\beta)\|z_{S'}\|$ and $\|b_z\| = \sin(\beta)\|z_{S'}\|$. It remains to study the singular values of the matrix

$$M_{\Theta,K} := \begin{pmatrix} -\rho & 0 \\ -(1+\rho)\|b_z\|/\|h_z\| & 1 \end{pmatrix} = \begin{pmatrix} -\rho & 0 \\ -(1+\rho)\tan(\beta) & 1 \end{pmatrix}.$$

Its two singular values are:

$$\sigma_{\max}(\Theta, K) := \frac{1}{2} \sqrt{(1+\rho)^2 + (1+\rho)^2 \tan(\beta)^2} + \frac{1}{2} \sqrt{(1-\rho)^2 + (1+\rho)^2 \tan(\beta)^2}, \quad (59)$$

$$\sigma_{\min}(\Theta, K) := \frac{1}{2} \sqrt{(1+\rho)^2 + (1+\rho)^2 \tan(\beta)^2} - \frac{1}{2} \sqrt{(1-\rho)^2 + (1+\rho)^2 \tan(\beta)^2}. \quad (60)$$

The singular values $\sigma_{\max}(\Theta, K) \ge 1$ and $\sigma_{\min}(\Theta, K) \le 1$ are monotonically increasing and decreasing, respectively, in $\tan(\beta)$. All other singular values of $\mathbf{J}\Theta|_K$ equal 1, hence (59) and (60) must the extremal maximal singular values of $\mathbf{J}\Theta|_K$. It is evident that $\rho = \text{vol}(K^c)/\text{vol}(K)$.
We develop explicit bounds for these singular values. The definition of the tangent and the decomposition $z_{S'} = h_z + b_z$ imply that $\tan(\beta) = \|b_z\|/\|h_z\|$.

Recall that $K \in \mathcal{K}$ is obtained from $T$ via barycentric subdivisions: first at $z_S$, leading to some intermediate $n$-simplex $T'$, and then at $z_{S'}$, leading to $K \subseteq T'$. Let $F_z \subseteq K$ be the face opposite to the vertex $z_{S'}$, which is some face of $T'$. Now, the height of $F_z$ in $K$ is $(n-\ell)$-th of the height of $F_z$ in $T'$ since $S'$ has dimension $n-\ell-1$. The height of the face $F_z$ in $T'$ is just the height of its opposing vertex, which lies in $S'$, and which equals the height of that vertex in $T$. Thus,

$$ \frac{\|b_z\|}{\|h_z\|} = \frac{\|b_z\|}{(n-\ell)^{-1}\|(n-\ell)h_z\|} \le \frac{\|z_{S'}\|}{(n-\ell)^{-1}\|(n-\ell)h_z\|} \le (n-\ell)\kappa_A(T). $$

We abbreviate

$$ \mu_{T,\ell} := \frac{1}{2} \sqrt{(1+\rho)^2 + (1+\rho)^2(n-\ell)^2 \kappa_A(T)^2 + \frac{1}{2} \sqrt{(1-\rho)^2 + (1+\rho)^2(n-\ell)^2 \kappa_A(T)^2}}. \quad (61) $$

This establishes bounds on the singular values of the transformation. In summary, the singular values of the Jacobian of $\Theta$ at almost every $x$ satisfy

$$ \sigma_1(\Theta, x) \le \mu_{T,\ell}, \quad \sigma_2(\Theta, x) = \dots = \sigma_{n-1}(\Theta) = 1, \quad \sigma_n(\Theta, x)^{-1} = \frac{\sigma_1(\Theta, x)}{|\det \mathbf{J}\Theta|_K} \le \mu_{T,\ell}/\rho. \quad (62) $$

Finally, we obtain $\Xi_1$ by extending $\Theta$ to all of the patch as the identity.

Having shown all the desired estimates, and the proof is complete. $\square$

**Remark 8.4.** We notice that $\Xi_1$ above is not only bi-Lipschitz, but in fact also piecewise affine with respect to an essentially non-overlapping simplicial decomposition of its domain. Whilst the reflection mapping in Proposition 8.3 serves our purpose, it might be possible to improve the analysis or construction and lower the Jacobian estimates.

**Remark 8.5.** The estimates in Proposition 8.3 are by no means optimal. Some improvements are immediately possible if $\ell = n-1$, corresponding to patches around faces. Then the reflection $\Xi$ in Lemma 3.4 satisfies the same properties as the mapping $\Xi_1$, mapping the simplex $T$ bijectively onto the $U$ and being the identity along the common face $S$. However, at any point, the singular values $\sigma_1 \ge \dots \ge \sigma_n$ of the Jacobian of $\Xi$ satisfy

$$ \begin{gathered} |\det \mathbf{J}\Xi| \le C_{\rho}(T), \\ \sigma_1, \sigma_n^{-1} \le \frac{1}{2} \sqrt{(C_{\theta}(T)\kappa_A(T) + 1)^2 + \kappa_A(T)^2 + \frac{1}{2} \sqrt{(C_{\theta}(T)\kappa_A(T) - 1)^2 + \kappa_A(T)^2}}, \\ \sigma_2, \dots, \sigma_{n-1}, \sigma_n \le 1. \end{gathered} $$

In particular,

$$ \sigma_1 \cdots \sigma_k |\det \mathbf{J}\Xi_2|^{-\frac{1}{p}} \le \left( \frac{1}{2} \sqrt{(C_{\theta}(T)\kappa_A(T) + 1)^2 + \kappa_A(T)^2} + \frac{1}{2} \sqrt{(C_{\theta}(T)\kappa_A(T) - 1)^2 + \kappa_A(T)^2} \right) C_{\rho}(T)^{\frac{1}{p}}. $$

These estimates improve over the ones in Proposition 8.3 when the reflection is over a single face. We believe that better estimates can be computed from the geometric data also in the other cases.

# 9 Constructive estimates of Poincaré-Friedrichs constants

We are now in the position to develop Poincaré-Friedrichs constants for the exterior derivative over domains with shellable triangulations; this includes the curl and divergence operators in three dimensions.

Our estimates use local Poincaré-Friedrichs constants over simplices with various boundary conditions. The analysis of the Poincaré potential operators in Section 6 has produced Poincaré-Friedrichs constants for bounded convex domains and their analogs for the case of full boundary conditions. Now, we also prove a Poincaré-Friedrichs inequality for differential forms over a simplex but subject to homogeneous boundary conditions along a collection of faces.
**Lemma 9.1.** Let $T$ be an $n$-simplex and let $\Gamma = F_0 \cup \cdots \cup F_\ell$ be the union of $\ell+1$ different faces ($\ell < n$) of $T$. Suppose that $u \in W^p\Lambda^k(T)$ such that $\text{tr}_{T,\Gamma} u = 0$. Then there exists $w \in W^p\Lambda^k(T)$ such that $\text{tr}_{T,\Gamma} w = 0$ and

$$dw = du, \quad \|w\|_{L^p(\Omega)} \le C_{\text{PF},T,\Gamma,\ell,k,p} \|du\|_{L^p(\Omega)}.$$

Here, $C_{\text{PF},T,\Gamma,\ell,k,p} > 0$ is a constant such that

$$C_{\text{PF},T,\Gamma,\ell,k,p} \le n! \cdot 2^{\ell+1} C_B(n, k+1) \text{vol}_{n-1}(S_1(0)) \cdot \kappa_M(T)^{k-1} C_{1,n}(T) \delta(T).$$

*Proof.* There exists an affine bijection $\varphi : \Delta^n \to T$ that maps the convex closure of the $n$ unit vectors onto the face $F_0$. We can also assume that the face of $\Delta^n$ orthogonal to the $i$-th coordinate axis, $1 \le i \le n$ is mapped onto the face $F_i$. In what follows, we let $\tilde{U}$ be the convex set obtained from reflecting $\Delta^n$ along the coordinate axes $\ell+1$ through $n$. We see that $\text{vol}(\tilde{U}) = 2^{n-\ell} \text{vol}(\Delta^n) = 2^{n-\ell}/(n!)$

We let $\hat{u} := \varphi^*u \in W^p\Lambda^k(\Delta^n)$ and define $\hat{g} \in L^p\Lambda^{k+1}(\Delta^n)$ via $\hat{g} := d\varphi^*u$. Then $d\hat{u} = \hat{g}$. We let $\tilde{u} \in W^p\Lambda^k(\tilde{U})$ be the extension of $\hat{u}$ onto $\tilde{U}$ by reflection across the coordinate axes, and let $\tilde{g} \in L^p\Lambda^{k+1}(\tilde{U})$ be the extension of $\hat{g}$ onto $\tilde{U}$ by reflection across the coordinate axes. By construction, $\tilde{u} \in W_0^p\Lambda^k(\tilde{U})$ with $\tilde{g} = d\tilde{u}$. We observe

$$\|\tilde{g}\|_{L^p(\tilde{U})} \le 2^{\frac{n-\ell}{p}} \|\hat{g}\|_{L^p(\Delta^n)}.$$

By the analysis for the Bogovskiï potential operators, Theorem 6.2, there exists $\hat{w} \in W_0^p\Lambda^k(\tilde{U})$ with $dw = d\tilde{u}$ satisfying the bounds

$$\|\hat{w}\|_{L^p(\tilde{U})} \le C_{\text{PF},\tilde{B},\tilde{U},k,p} \|\tilde{g}\|_{L^p(\tilde{U})}. \qquad (63)$$

Here,

$$\begin{aligned} C_{\text{PF},\tilde{B},\tilde{U},k,p} &= C_{\tilde{B}}(n, k+1) \text{vol}_{n-1}(S_1(0)) \frac{\delta(\tilde{U})^n}{\text{vol}(\tilde{U})} \delta(\tilde{U}) \\ &= C_{\tilde{B}}(n, k+1) \text{vol}_{n-1}(S_1(0)) \frac{2^n}{2^{n-\ell/n!}} 2 = C_{\tilde{B}}(n, k+1) \text{vol}_{n-1}(S_1(0)) \cdot n! 2^{\ell+1}. \end{aligned}$$

We let $\tilde{w} \in W^p\Lambda^k(\tilde{U})$ be defined by averaging the $2^{n-\ell}$ reflections of $\hat{w}$ along the coordinate axes $\ell+1$ through $n$. Thus,

$$\|\tilde{w}\|_{L^p(\Delta^n)} = 2^{-\frac{n-\ell}{p}} \|\tilde{w}\|_{L^p(\tilde{U})}, \quad \|\tilde{w}\|_{L^p(\tilde{U})} \le \|\hat{w}\|_{L^p(\tilde{U})}.$$

Since $\tilde{g}$ has the same symmetries, $dw = d\tilde{u} = \tilde{g}$.

We let $w \in W^p\Lambda^k(T)$ be defined by $w := \varphi^{-*}(\tilde{w}|_{\Delta^n})$. By construction, $dw = du$, and $w$ has vanishing trace along $F_0 \cup \cdots \cup F_\ell$.

To complete the discussion, we recall by Proposition 5.3 that

$$\begin{aligned} \|w\|_{L^p(T)} &\le |\det(\mathbf{J}\varphi)|^{\frac{1}{p}} \|\mathbf{J}\varphi^{-1}\|_2^k \|\hat{w}\|_{L^p(\Delta^n)}, \\ \|\hat{g}\|_{L^p(\Delta^n)} &\le |\det(\mathbf{J}\varphi^{-1})|^{\frac{1}{p}} \|\mathbf{J}\varphi\|_2^{k+1} \|du\|_{L^p(T)}. \end{aligned}$$

It remains to use Lemma 3.3, and the desired inequality is shown. $\square$

**Remark 9.2.** Under the assumptions of Lemma 9.1, the special case $p = 2$ may allow for a better estimate for the auxiliary constant in (63). We conjecture the Poincaré-Friedrichs constants over bounded convex domains are non-increasing in the form degree $k$, as is already known for smoothly bounded convex domains [39]. The complete inequality then reads: there exists $w \in W^2\Lambda^k(T)$ such that $\text{tr}_{T,\Gamma} w = 0$ and

$$dw = du, \quad \|w\|_{L^2(\Omega)} \le C_{\text{PF},T,\Gamma,\ell,k,2} \|du\|_{L^2(\Omega)},$$

where

$$C_{\text{PF},T,\Gamma,\ell,k,2} \le \frac{2}{\pi} \kappa_M(T)^{k-1} C_{1,n}(T) \delta(T).$$

We conjecture that the constant can be improved further, to be independent of the tetrahedron's eccentricity, and that the result can be extended to Lebesgue exponents $1 \le p \le \infty$.
We prepare some further notation for our two main results. Whenever $\mathcal{T}$ is an $n$-dimensional triangulation, we use the abbreviation

$$C_{\mathrm{PF},\Gamma,k,p}(\mathcal{T}) := \max_{\substack{T \in \mathcal{T} \\ \dim(T)=n \\ 0 \le l \le n}} C_{\mathrm{PF},T,\Gamma,l,k,p}.$$

Suppose that $\mathcal{T}$ is an $n$-dimensional triangulation and has a shelling $T_0, T_1, \dots, T_M$. For each $0 \le m \le M$, we write

$$X_m := T_0 \cup T_1 \cup \dots \cup T_m.$$

By Lemma 7.7, each $X_m$ is a triangulated $n$-dimensional submanifold with boundary. For each $1 \le m \le M$, the interface of the new simplex to the previous intermediate triangulation is

$$\Gamma_m := T_m \cap X_{m-1}.$$

According to Lemma 7.8, for each $1 \le m \le M$, there exists an interior simplex $S_m \in \mathcal{T}$ such that $T_m$ is the last $n$-simplex in the shelling that belongs to $\mathrm{st}_{\mathcal{T}}(S_m)$. In particular, already $\mathrm{st}_{\mathcal{T}}(S_m) \subseteq X_m$.

In addition to that, we introduce

$$U_{m-1} := \mathrm{st}_{\mathcal{T}}(S_m) \setminus T_m$$

for the complement $U_{m-1} \subseteq X_{m-1}$ of the simplex $T_m$ in the star $\mathrm{st}_{\mathcal{T}}(S_m)$.

This manuscript's two main results for shellable triangulations of domains, not necessarily convex or even star-shaped, are the following theorems. The first one (Theorem 9.3) is inspired by the recursive estimate in Theorem 4.4: we always construct a potential over the new simplex using the trace value already known along the connecting faces, thereby constructing a potential over larger and larger subdomains and ensuring the correct continuity conditions. The second one (Theorem 9.4) then unfolds the recursive estimates into the final Poincaré-Friedrichs inequality.

**Theorem 9.3.** Let $\mathcal{T}$ be a shellable $n$-dimensional triangulation, and let the domain $\Omega \subseteq \mathbb{R}^n$ be the interior of the underlying set of $\mathcal{T}$. Let $T_0, T_1, \dots, T_M$ be a shelling of $\mathcal{T}$. Then for any $u \in W^p \Lambda^k(\Omega)$, where $1 \le p \le \infty$, there exists $w \in W^p \Lambda^k(\Omega)$ with $dw = du$ and such that the following estimates hold:

$$\|w\|_{L^p(T_0)} \le C_{\mathrm{PF},\mathcal{P},T_0,k,p} \|du\|_{L^p(T_0)},$$

and for each $T_m \in \mathcal{T}$, $1 \le m \le M$, we have the recursive estimate

$$\|w\|_{L^p(T_m)} \le C_{\mathrm{PF},T_m,\Gamma_m,k,p}(\mathcal{T}) (\|du\|_{L^p(T_m)} + C_{5,n,k+1,l,p}(\mathcal{T}) \|du\|_{L^p(U_{m-1})}) + C_{5,n,k,l,p}(\mathcal{T}) \|w\|_{L^p(U_{m-1})},$$
where $0 \le l < n$ is such that $T_m$ has $n-l$ faces in common with the previous simplices.

*Proof.* Let $u \in W^p \Lambda^k(\Omega)$. First, from Theorem 6.2, there exists $w_0 \in W^p \Lambda^k(T_0)$ satisfying $dw_0 = du$ over $T_0$ together with

$$\|w_0\|_{L^p(T_0)} \le C_{\mathrm{PF},\mathcal{P},T_0,k,p} \|du\|_{L^p(T_0)}.$$

Suppose that $0 < m \le M$ such that there exists $w_{m-1} \in W^p \Lambda^k(X_{m-1})$ with $dw_{m-1} = du$ over $X_{m-1}$. This is already true for $m = 1$. By assumption, $T_m$ and $X_{m-1}$ share the interface $\Gamma_m$, which is a collection of faces of $T_m$. In accordance to Lemma 7.8, adding $T_m$ completes a star in $\mathcal{T}$ around some simplex $S_m$, and we let $U_{m-1} \subseteq X_{m-1}$ be the complement of $T_m$ in that newly completed star. Write $l$ for the dimension of $S_m$.

Similar as in [25, Equations (5.12) and (5.14)] or [19, Equations (6.7) and (6.9)], we now define the field $\tilde{w}_m \in W^p \Lambda^k(T_m)$ via

$$\tilde{w}_m := u|_{T_m} + \Xi_1^* ((w_{m-1} - u)|_{U_{m-1}}). \qquad (64)$$

With $u \in W^p \Lambda^k(\Omega)$, Equation (44), and the mapping $\Xi_1$ of Proposition 8.3 being the identity along $\Gamma_m$, one verifies that

$$\begin{align*}
\mathrm{tr}_{\Gamma_m} \tilde{w}_m &= \mathrm{tr}_{\Gamma_m} u|_{T_m} + \mathrm{tr}_{\Gamma_m} \Xi_1^* w_{m-1|_{U_{m-1}}} - \mathrm{tr}_{\Gamma_m} \Xi_1^* u|_{U_{m-1}} \\
&= \mathrm{tr}_{\Gamma_m} \Xi_1^* w_{m-1|_{U_{m-1}}} \\
&= \mathrm{tr}_{\Gamma_m} w_{m-1|_{U_{m-1}}}.
\end{align*}$$
Moreover, again by (44)

$$ 
\begin{aligned}
d\tilde{w}_m &= du|_{T_m} + d\Xi_1^* w_{m-1|U_{m-1}} - d\Xi_1^* u|_{U_{m-1}} \\
&= du|_{T_m} + \Xi_1^* dw_{m-1|U_{m-1}} - \Xi_1^* du|_{U_{m-1}} = du|_{T_m},
\end{aligned}
$$

since $dw_{m-1} = du$ over $X_{m-1}$ and thus in particular over $U_{m-1}$. Setting $w_m := w_{m-1}$ over $X_{m-1}$ and $w_m := \tilde{w}_m + w_m''$ over $T_m$, we thus verify that $w_m \in W^p \Lambda^k(X_m)$ with $dw_m = du$ over $X_m$.
We now estimate norms. By construction,

$$ \|w_m\|_{L^p(T_m)} \le \|\Xi_1^* w_m|_{U_{m-1}}\|_{L^p(T_m)} + \|u - \Xi_1^* u|_{U_{m-1}}\|_{L^p(T_m)}. $$

Due to Lemma 9.1, which applies since $\text{tr}_{\Gamma_m}(u|_{T_m} - \Xi_1^* u|_{U_{m-1}}) = 0$, and again using (44) we have

$$ 
\begin{aligned}
\|u - \Xi_1^* u|_{U_{m-1}}\|_{L^p(T_m)} &\le C_{\text{PF},T_m,\Gamma_m,\ell,k,p} (\|du\|_{L^p(T_m)} + \|d\Xi_1^* u|_{U_{m-1}}\|_{L^p(T_m)}) \\
&= C_{\text{PF},T_m,\Gamma_m,\ell,k,p} (\|du\|_{L^p(T_m)} + \|\Xi_1^* du|_{U_{m-1}}\|_{L^p(T_m)}).
\end{aligned}
$$

Proposition 5.3 and Proposition 8.3 now show

$$ 
\begin{aligned}
\|\Xi_1^* w_m|_{U_{m-1}}\|_{L^p(T_m)} &\le C_{5,n,k,\ell,p}(T) \|w_m\|_{L^p(U_{m-1})}, \\
\|\Xi_1^* du|_{U_{m-1}}\|_{L^p(T_m)} &\le C_{5,n,k+1,\ell,p}(T) \|dw_m\|_{L^p(U_{m-1})}.
\end{aligned}
$$

We have assumed that $dw_{m-1} = du|_{X_{m-1}}$. The existence of $w \in W^p \Lambda^k(\Omega)$ satisfying the recursive estimate follows by setting $w|_{T_m} := w_m$ for all $0 \le m \le M$. $\square$

We finally derive an estimate for the Poincaré-Friedrichs constant, in analogy to Theorem 4.5. For ease of discussion, we choose to present the result for a general class of recursive estimates. Of course, implementations should make use of the simplifications warranted by the specific recursive structure, such as the recursion presented in Theorem 9.3.

**Theorem 9.4.** Let $\mathcal{T}$ be a shellable $n$-dimensional triangulation, and let the domain $\Omega \subseteq \mathbb{R}^n$ be the interior of the underlying set of $\mathcal{T}$. Let $T_0, T_1, \dots, T_M$ be a shelling of $\mathcal{T}$. Suppose that $u, w \in W^p \Lambda^k(\Omega)$, where $1 \le p \le \infty$, such that $dw = du$ and the following recursive estimate holds:

$$ \|w\|_{L^p(T_0)} \le A_{0,0} \|du\|_{L^p(T_0)}, $$

and for each $T_m \in \mathcal{T}$, $1 \le m \le M$, we have the recursive estimate

$$ \|w\|_{L^p(T_m)} \le \sum_{\ell=0}^{m} A_{\ell} \|du\|_{L^p(T_{\ell})} + \sum_{\ell=0}^{m-1} B_{m,\ell} \|w\|_{L^p(T_{\ell})}. $$

Then we have an inequality of the form

$$ \|w\|_{L^p(\Omega)} \le \left( \sum_{m=0}^{M} \left( \sum_{\ell=0}^{M} C_{m,\ell}^q \right)^{\frac{p}{q}} \right)^{\frac{1}{p}} \|du\|_{L^p(\Omega)}, $$

where $q \in [1, \infty]$ satisfies $1 = 1/p + 1/q$ and with obvious modifications if $p = 1$ or $p = \infty$. Here,

$$ C_{m,\ell} = \sum_{m=i_L > \dots > i_1 \ge \ell} B_{i_L, i_{L-1}} \cdots B_{i_2, i_1} A_{i_1, \ell}. $$

*Proof.* Unwrapping the recursion, we obtain an estimate of the form

$$ \|w\|_{L^p(T_m)} \le \sum_{\ell=0}^{m} \underbrace{\left( \sum_{m=i_L > \dots > i_1 \ge \ell} B_{i_L, i_{L-1}} \cdots B_{i_2, i_1} A_{i_1, \ell} \right)^{\frac{p}{q}}}_{=: C_{m,\ell}} \|du\|_{L^p(T_\ell)} $$
Here, $C_{m,\ell}$ denotes the coefficient of $\|du\|_{L^p(T_\ell)}$, possibly zero, as it appears in the unwrapped recursive estimate of $\|w\|_{L^p(T_m)}$. Once again, The global Poincaré-Friedrichs inequality follows via Hölder's inequality:

$$ \|w\|_{L^p(\Omega)}^p = \sum_{m=0}^{M} \|w\|_{L^p(T_m)}^p \le \sum_{m=0}^{M} \left( \sum_{\ell=0}^{M} C_{m,\ell}^q \right)^{\frac{p}{q}} \sum_{\ell'=0}^{M} \|du\|_{L^p(T_{\ell'})}^p \le \left( \sum_{m=0}^{M} \left( \sum_{\ell=0}^{M} C_{m,\ell}^q \right)^{\frac{p}{q}} \right) \|du\|_{L^p(\Omega)}^p $$

where $q \in [1, \infty]$ is as described above. The proof is complete. □

# 10 Numerical examples

We wish to assess the practical quality of our upper bounds, with exclusive focus on the Hilbert space case $p = 2$. To that end, we compare our estimates for the Poincaré-Friedrichs constants with the exact constants of a few examples domains in dimension 2 and 3. In lieu of the exact values, finite element eigenvalue computations on a refined mesh provide reference values as a proxy for the exact value.

Theorem 9.4 estimates the gradient, curl and divergence Poincaré-Friedrichs constants. In addition, Theorem 4.5 provides another upper bound for the gradient Poincaré-Friedrichs constant. We remark that the Poincaré-Friedrichs constant for the divergence can always be estimated by less than the diameter of the domain (see Lemma 4.1).

## 10.1 Software and algorithms

All computations were implemented with an in-house C++ code implementing finite element exterior calculus. The reference Poincaré constants are the inverse square roots of reference eigenvalues, which are computed as solutions of finite element eigenvalue problems in mixed formulation over sufficiently refined meshes.

To compute the upper bound for the Poincaré constant of Theorem 4.5, we need to find paths between simplices. One quickly sees that a spanning tree of the face-connection graph provides paths that minimize the estimate. We choose a depth-first search that attempts to minimize the resulting constant. While this is feasible for small two-dimensional examples, the computation takes considerably longer for our three-dimensional examples.

We need a shelling to apply Theorem 4.5 and to estimate the Poincaré-Friedrichs constant. Finding a shelling optimizing a geometric target quantity is a challenging problem in computational geometry and theoretical computer science. Brute-force enumeration of shellings is still feasible for small two-dimensional triangulations but becomes practically infeasible in dimension three because the number of shellings sees a combinatorial explosion. Instead, we employ a backtracking search guided by a greedy strategy: Starting with a single simplex, we successively add simplices to the partial shellings until a complete shelling is found. Among the candidates for the next simplex, our practical heuristic prefers those that introduce the smallest possible geometric constants. We always prioritize the completion of face stars because then the induced constants are easier to control via Lemma 3.4. Repeating this backtracking search for all possible initial simplices and generating up to 10 shellings each time, we pick the optimal shelling among those found. From our computations, we have observed that the Poincaré-Friedrichs estimates from different shellings may differ by orders of magnitudes; that said, the method may not necessarily produce the global optimum.

In order to ensure tight estimates, we compute the relevant geometric parameters, such as heights, individually for each simplex and use individualized estimates for each local star. These estimates are numerically tighter than the ones stated in Theorems 4.5 and 9.4, as we directly implement the calculations used in the proofs.

## 10.2 Estimates for partial boundary conditions

The Poincaré-Friedrichs constants over simplices subject to boundary conditions along a few faces frequently enter our estimates. We therefore prefer to have tight upper estimates for them.

Whenever $T$ is a simplex, the Poincaré-Friedrichs constants have upper bounds of the form $C\delta(T)$ for some numerical constant $C > 0$. Lemma 9.1 and the subsequent remark (using $p = 2$) have established
<table><caption>Table 1: Reference $L^2$ Poincaré–Friedrichs constants for the gradient, curl, and divergence operators over the reference tetrahedron, where boundary conditions are imposed along the boundary part $\Gamma$, computed via lowest-order finite element methods on a refined mesh. Notably, the top-left should and the bottom-right should asymptotically coincide, as should the bottom-left and the top-right. By comparison, Lemma 9.1 and Remark 9.2 give the upper bound $2/\pi \cdot \delta(\Delta^3) \approx 0.90031631615$.</caption><thead><tr><th></th><th>grad</th><th>curl</th><th>div</th></tr></thead><tbody><tr><td>Γ = F<sub>0</sub> ∪ F<sub>1</sub> ∪ F<sub>2</sub> ∪ F<sub>3</sub></td><td>0.0862501765</td><td>0.1453729386</td><td>0.2601720480</td></tr><tr><td>Γ = F<sub>1</sub> ∪ F<sub>2</sub> ∪ F<sub>3</sub></td><td>0.1093817645</td><td>0.1829680131</td><td>0.3493931507</td></tr><tr><td>Γ = F<sub>2</sub> ∪ F<sub>3</sub></td><td>0.1450219664</td><td>0.2428248005</td><td>0.1845090722</td></tr><tr><td>Γ = F<sub>3</sub></td><td>0.2057601732</td><td>0.1527746636</td><td>0.1223402289</td></tr><tr><td>Γ = ∅</td><td>0.2631059409</td><td>0.1458215887</td><td>0.0874066554</td></tr></tbody></table>

that $C = 2/\pi \approx 0.45015815807$. In the special case of the gradient potential, where the Poincaré constant can be characterized variationally, the Poincaré constant without boundary conditions is already an upper bound for the Poincaré constant with boundary conditions, and one can choose $C = 1/\pi \approx 0.31830988618$. This conforms to the well-known fact that the eigenvalues of the scalar Laplacian with mixed boundary conditions are between the Neumann and Dirichlet eigenvalues. Finally, the factor $1/\pi$ in the previous two estimates has a known improvement when the simplex is a triangle: then we can replace the factor $1/\pi$ by $1/j_{1,1} \approx 0.2609803592$, where $j_{1,1}$ is the zero of a Bessel function; see [45].

For the purpose of comparison, we compute reference Poincaré–Friedrichs constants for $p = 2$ over the reference tetrahedron via a higher-order finite element method over a mesh after three iterations of uniform refinement. These data are tabulated in Table 1. The upper bound provided by Lemma 9.1 and Remark 9.2 has an overestimate by a factor of at most six. The estimate seems within practical range, at least on simplices with good shape regularity. We expect the overestimate to worsen as the eccentricity of the tetrahedron increases. The results indicate room for improvement and future research should tighten the gap towards the practically observed values, including for ill-shaped simplices.

## 10.3 Two-dimensional examples

We consider the following example domains in two dimensions: the unit square $\Omega_Q$, the L-shaped domain $\Omega_L$, and the slit domain $\Omega_S$:

$$ \Omega_Q = (0, 1)^2, \quad \Omega_L = (-1, 1)^2 \setminus [0, 1)^2, \\ \Omega_S = (-1, 1)^2 \setminus ([0, 1) \times \{0\}). $$

We consider the following triangulations; see also Figure 7:

* $\mathcal{T}_Q$: square triangulation with two triangles

* $\mathcal{T}_L$: L-shaped domain triangulation with four triangles

* $\mathcal{T}_S$: Slit domain triangulation, five triangles

* $\mathcal{T}_{S'}$: Slit domain triangulation, 8 triangles, all of which touch the origin.

* $\mathcal{T}_{S''}$: Slit domain triangulation, 8 triangles, four of which touch the origin.

The de Rham complex of a planar domain has only two differential operators, and their Poincaré–Friedrichs constants are the inverse square roots of the smallest non-zero Dirichlet and Neumann Laplacians. Standard finite element eigenvalue computations on any sufficiently refined mesh provide acceptable reference values for these; we choose four steps of uniform refinement.

We compare the reference Poincaré constants with the gradient potential estimate in Theorem 4.5. Moreover, we compare the reference Poincaré–Friedrichs constants with the bounds computed in Theorem 9.4. Here, we take into account Lemma 3.4 and Proposition 8.3.

The shelling-based estimate for the Poincaré constant of the gradient potential outperforms the estimate in Section 4 in our examples where the triangles are not all congruent to each other ($\mathcal{T}_L$ and
<figure><img src="image_11.png" /><figcaption>Table 2: Estimates for $L^2$ Poincaré–Friedrichs constants over various triangulated 2D domains. Reference values for the gradient and divergence (2nd and 7th column) computed with finite element methods together with estimates and ratios: Theorem 4.5 (3rd and 4th column), using Theorem 9.4 with $k = 0$ (5th and 6th column), and using Theorem 9.4 with $k = 1$ (8th and 9th column). Note that the divergence constant can always be estimated using Lemma 4.1.</figcaption></figure>

<table><thead><tr><th></th><th>grad ref</th><th>grad est</th><th>grad ratio</th><th>grad est</th><th>grad ratio</th><th>div ref</th><th>div est</th><th>div ratio</th></tr></thead><tbody><tr><td>T<sub>Q</sub></td><td>0.318</td><td>0.904</td><td>2.842</td><td>0.904</td><td>2.842</td><td>0.225</td><td>1.339</td><td>5.953</td></tr><tr><td>T<sub>L</sub></td><td>0.822</td><td>2.381</td><td>2.896</td><td>1.421</td><td>1.729</td><td>0.322</td><td>2.505</td><td>7.781</td></tr><tr><td>T<sub>L'</sub></td><td>0.822</td><td>2.391</td><td>2.909</td><td>2.391</td><td>2.909</td><td>0.322</td><td>4.053</td><td>12.587</td></tr><tr><td>T<sub>S</sub></td><td>0.978</td><td>2.752</td><td>2.814</td><td>1.779</td><td>1.819</td><td>0.346</td><td>3.717</td><td>10.744</td></tr><tr><td>T<sub>S'</sub></td><td>0.978</td><td>3.131</td><td>3.202</td><td>3.131</td><td>3.202</td><td>0.346</td><td>5.453</td><td>15.762</td></tr><tr><td>T<sub>S''</sub></td><td>0.978</td><td>2.761</td><td>2.824</td><td>2.761</td><td>2.824</td><td>0.346</td><td>4.662</td><td>13.476</td></tr></tbody></table>

$T_{S,5}$). That is a result of our implementation, where the face-based reflections are not necessarily onto, which allows for some improved estimates. The results are summarized in Table 2. The maximal overestimation factor is below 16, which we deem reasonable.

## 10.4 Three-dimensional examples

We consider the following example domains in three dimensions: the unit cube $\Omega_C$, the Fichera corner domain $\Omega_F$, and the crossed bricks domain $\Omega_B$:

$$ \begin{aligned} \Omega_C &= (0, 1)^3, & \Omega_F &= (-1, 1)^3 \setminus [0, 1)^3, \\ \Omega_B &= ((-1, 0) \times (-1, 0) \times (-1, 1)) \\ & \cup ((-1, 0) \times (-1, 1) \times (-1, 0)) \\ & \cup ((-1, 1) \times (-1, 0) \times (-1, 0)). \end{aligned} $$

The following triangulations are used in computations:

* $T_{C,5}$: cube triangulation with five tetrahedra

* $T_{C,K}$: Kuhn triangulation of the cube, consisting of six tetrahedra

* $T_{B,5}$: crossed bricks, four copies of $T_{C,5}$

* $T_{B,K}$: crossed bricks, four copies of $T_{C,K}$

* $T_F$: Fichera corner, 24 simplices

The reference Poincaré–Friedrichs constants for the gradient, curl, and divergence operators are found via standard finite element eigenvalue computations, using the lowest-order finite element de Rham complex over a sufficiently refined mesh (four steps of uniform refinement). Again, we compare these reference values with estimates obtained Theorem 9.4, and in the special case of the gradient, with Theorem 4.5. The results, summarized in Tables 3 and 4, are less satisfactory than in the two-dimensional case; we deem the overestimation factors as still reasonable for patches with few tetrahedra, but they are very high for the Fichera corner with 24 simplices.
<table><caption>Table 3: Estimates for $L^2$ Poincaré-Friedrichs constants of the gradient over various triangulated 3D domains. Reference values for the gradient (2nd column) computed with finite element methods together with estimates and ratios: Theorem 4.5 (3rd and 4th column), and using Theorem 9.4 with $k = 0$ (5th and 6th column).</caption><thead><tr><th></th><th>grad ref</th><th>grad est</th><th>grad ratio</th><th>grad est</th><th>grad ratio</th></tr></thead><tbody><tr><td>T<sub>C,5</sub></td><td>0.317</td><td>4.317</td><td>13.581</td><td>3.246</td><td>10.214</td></tr><tr><td>T<sub>C,K</sub></td><td>0.317</td><td>3.571</td><td>11.23</td><td>11.034</td><td>34.722</td></tr><tr><td>T<sub>B,5</sub></td><td>1.022</td><td>7.698</td><td>7.5285</td><td>25.972</td><td>25.397</td></tr><tr><td>T<sub>B,K</sub></td><td>1.022</td><td>10.106</td><td>9.8824</td><td>53.763</td><td>52.573</td></tr><tr><td>T<sub>F</sub></td><td>0.711</td><td>10.622</td><td>14.927</td><td>268.071</td><td>377.033</td></tr></tbody></table>

<table><caption>Table 4: Estimates for $L^2$ Poincaré-Friedrichs constants over various triangulated 3D domains. Reference values for the curl and divergence (2nd and 5th column) computed with finite element methods together with estimates and ratios that rely on Theorem 9.4 with $k = 1$ (3rd and 4th column) and with $k = 2$ (5th and 6th column). Note that the divergence constant can always be estimated using Lemma 4.1.</caption><thead><tr><th></th><th>curl ref</th><th>curl est</th><th>curl ratio</th><th>div ref</th><th>div est</th><th>div ratio</th></tr></thead><tbody><tr><td>T<sub>C,5</sub></td><td>0.225</td><td>141.148</td><td>627.310</td><td>0.183</td><td>3.391</td><td>18.453</td></tr><tr><td>T<sub>C,K</sub></td><td>0.225</td><td>12.157</td><td>54.030</td><td>0.183</td><td>25.899</td><td>140.920</td></tr><tr><td>T<sub>B,5</sub></td><td>0.331</td><td>113.084</td><td>341.056</td><td>0.233</td><td>152.886</td><td>655.993</td></tr><tr><td>T<sub>B,K</sub></td><td>0.331</td><td>162.687</td><td>490.655</td><td>0.233</td><td>512.273</td><td>2198.026</td></tr><tr><td>T<sub>F</sub></td><td>0.554</td><td>25752.342</td><td>46403.302</td><td>0.310</td><td>8958.467</td><td>28835.266</td></tr></tbody></table>

# 11 Outlook

This manuscript contributes upper bounds for Poincaré-Friedrichs constants. The main application is the curl operator over local patches in low spatial dimensions (i.e., $n = 2$ or $n = 3$). However, we believe there are ample opportunities to refine our estimates of Poincaré-Friedrichs constants, both conceptually and algorithmically, and extend the technique to further applications.

We use local Poincaré-Friedrichs inequalities over single simplices, subject to boundary conditions along some faces. We believe that these inequalities can be tightened. In particular, we conjecture that the improvements for triangles [45] have analogs for tetrahedra.

The gradient Poincaré-Friedrichs constant already dominates the Poincaré-Friedrichs constants of the $L^2$ de Rham complex without boundary conditions (or with full boundary conditions) over bounded smooth convex domains [39]. We believe that this relationship extends to the entire range of Lebesgue exponents $1 \le p \le \infty$ over bounded convex domains; this would improve on the Poincaré-Friedrichs constants derived via the regularized Poincaré and Bogovskii operators.

Our method hinges on extending differential forms onto a simplex from the complement of that simplex within a local star. The present manuscript realizes this extension via pullback along a bi-Lipschitz mapping. It seems worthwhile to explore other techniques, e.g., carefully analyzing trace and extension theorems.

On the algorithmic part, merely checking whether a general triangulation has a shelling is known to be an NP-complete problem and hence computationally infeasible in practice [35]. Nevertheless, there might be more efficient algorithms for constructing shellings for practically relevant triangulations while optimizing geometric target heuristics.

We anticipate the present results to generalize to shellable polytopal complexes. Apart from the intrinsic interest in polytopal complexes, such extensions are already relevant for simplicial triangulations: As a rule of thumb, the estimates deteriorate as the number of simplices increases. Lumping simplices into a few polytopal subdomains for which the Poincaré-Friedrichs constants are controllable could improve the estimates. For example, the crossed bricks domain easily partitions into four quadrilaterals.

Lastly, estimating Poincaré-Friedrichs constants over domains and manifolds with shellable triangulations is fundamentally restricted to topological balls and spheres. However, we expect our estimates over local patches to serve as subcomponents in estimating Poincaré-Friedrichs constants of general