*p*-robust equivalence of global continuous and local discontinuous approximation, a *p*-stable local projector, and optimal elementwise *hp* approximation estimates in *H*¹

Martin Vohralík

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Martin Vohralík. *p*-robust equivalence of global continuous and local discontinuous approximation, a *p*-stable local projector, and optimal elementwise *hp* approximation estimates in *H*¹. 2024. hal-04436063v2

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p-robust equivalence of global continuous and local discontinuous approximation, a p-stable local projector, and optimal elementwise hp approximation estimates in H 1

Martin Vohralík‡§

July 7, 2025

Abstract

Consider an open bounded connected Lipschitz polygon or polyhedron Ω, a function v in the Sobolev
space H¹(Ω), and a simplicial mesh T_h of Ω. We prove the equivalence of two piecewise p-polynomial
best approximations of v in the H¹ seminorm: 1) globally on the whole domain Ω, with the (trace)
continuity requirement; 2) locally on each mesh element, without any interelement continuity re-
quirement. The former (global continuous) approximation arises in numerical methods for partial
differential equations related to the H¹(Ω) space, whereas the latter (local discontinuous) approxi-
mation is a key quantity in approximation theory. Crucially, we establish p-robustness in that the
equivalence constant only depends on the mesh shape regularity and the spatial dimension. This
improves the recent results of [Found. Comput. Math. 16 (2016), 723–750] and [Numer. Math. 135
(2017), 1073–1119], where the equivalence constant was possibly dependent (algebraically or logarith-
mically) on the underlying polynomial degree. Consequently, we obtain fully h- and p- (mesh-size-
and polynomial-degree-) optimal approximation estimates under the minimal Sobolev regularity only
requested separately on each mesh element. These two results immediately follow by our construction
of an operator from the infinite-dimensional Sobolev space H¹(Ω) to its finite-dimensional piecewise
polynomial subspace that has simultaneously all the following properties: 1) it is defined over the en-
tire H¹(Ω) and preserves boundary conditions imposed on a part of the boundary of Ω; 2) it is defined
locally in a neighborhood of each mesh element; 3) the only information about the given function
it uses is its elementwise H¹-orthogonal polynomial projection; 4) it is a projector, i.e., it preserves
continuous piecewise polynomials; 5) it is locally and p-robustly stable in the H¹(Ω)-seminorm; 6)
its approximation property is locally and p-robustly equivalent to that of the local discontinuous
 elementwise H¹-orthogonal) projection.

**Key words:** Sobolev space $H^1$, best approximation, continuous approximation, discontinuous approximation, piecewise polynomial, local–global equivalence, minimal regularity, elementwise regularity, projector, *hp* finite elements, error bound, polynomial-degree robustness

# 1 Introduction

Let $\Omega \subset \mathbb{R}^d$, $d = 2, 3$, be an open, bounded, and connected Lipschitz polygon or polyhedron and $H^1(\Omega)$
the Sobolev space of functions square-integrable together with their weak gradients, cf. Ciarlet [14] or
Ern and Guermond [20]. Let $\mathcal{T}_h$ be a simplicial mesh of $\Omega$.

1.1 h and hp approximation in H 1

The question of error arising in approximation of $v \in H^1(\Omega)$ by continuous piecewise polynomials over $\mathcal{T}_h$
is central in numerical approximation and namely in finite element analysis [14, 20]. Numerous (quasi-)
interpolation operators were proposed and studied in the past in Clément [15], Scott and Zhang [34],
Bernardi and Girault [7], Falk and Winther [24], Bank and Ovall [6], Ern and Guermond [19], Arnold
and Guzmán [3], Gawlik et al. [26], and the references therein. Often, the approximation error is studied
with respect to the mesh size $h$, for a bounded (and uniform) polynomial degree $p$. In the $hp$ context,

<footnote>‡Inria, 2 rue Simone Iff, 75589 Paris, France (martin.vohralik@inria.fr).</footnote>

<footnote>§CERMICS, Ecole des Ponts, 77455 Marne-la-Vallée, France</footnote>
the error with respect to both the principal parameters, the mesh size $h$ and the polynomial degree $p$, needs to be examined, see Babuška and Suri [5], Schwab [33], Demkowicz and Buffa [16], Melenk [30], Karkulik and Melenk [27], and the references therein.

## 1.2 Equivalence of global continuous and local discontinuous approximation

A central result has been recently established in Veeser [35], see also the predecessor results in Carstensen et al. [11, Theorem 2.1 and inequalities (3.2), (3.5), and (3.6)] and Aurada et al. [4, Proposition 3.1]. It shows the equivalence of the global-best approximation of $v$ by continuous piecewise polynomials over $\mathcal{T}_h$ with the local-best approximation of $v$ by discontinuous piecewise polynomials over $\mathcal{T}_h$ in the $H^1$ seminorm. This seems surprising at a first sight, since the discontinuous approximation space is (much) bigger than the continuous one. In [35, Theorem 2], this result is established for a bounded (and uniform) polynomial degree $p$, where the hidden equivalence constant degrades (algebraically) with increasing the polynomial degree. This comes both from the construction, passing through the Lagrange degrees of freedom, and the tools, employing an inverse inequality. Improvement of this dependence (to logarithmic) in two space dimensions has been later developed in Canuto et al. [10, Theorem 4] in the context of study of convergence and optimality $hp$ finite elements. Here, the logarithmic dependence on $p$ comes from the use of two-dimensional almost optimal polynomial extension operators.

## 1.3 Main results of this manuscript

Our main result is a construction of an interpolation operator from $H_{0,D}^1(\Omega)$ to continuous piecewise polynomials over $\mathcal{T}_h$, $P_{hp}: H_{0,D}^1(\Omega) \to \mathcal{P}_p(\mathcal{T}_h) \cap H_{0,D}^1(\Omega)$ (details on notation are given in Section 2 below) with the following properties: 1) it is defined over the entire $H_{0,D}^1(\Omega)$, preserving homogeneous boundary conditions on a part $\Gamma_D$ of the boundary of $\Omega$; 2) it is defined locally in a neighborhood of each mesh element $K \in \mathcal{T}_h$; 3) the only information about the given function it uses is its elementwise $H^1$-orthogonal polynomial projection; 4) it is a projector, i.e., it preserves continuous piecewise polynomials,

$$ P_{hp}(v) = v \quad \forall v \in \mathcal{P}_p(\mathcal{T}_h) \cap H_{0,D}^1(\Omega); \quad (1.1) $$
(projection)

5) it is locally and $p$-robustly stable in the $H^1(\Omega)$-seminorm, i.e.,

$$ \|\nabla P_{hp}(v)\|_K^2 \lesssim \sum_{L \in \bar{\mathcal{T}}_K} \|\nabla v\|_L^2 \quad \forall v \in H_{0,D}^1(\Omega), \quad (1.2) $$
(local $H^1$-seminorm stability)

where $\tilde{\mathcal{T}}_K$ is an extended element patch consisting of two layers of vertex neighbors of $K \in \mathcal{T}_h$; 6) its approximation property is locally and $p$-robustly equivalent to that of the local discontinuous (elementwise $H^1$-orthogonal) projection:

$$ \|\nabla(v - P_{hp}(v))\|_K^2 \lesssim \sum_{L \in \bar{\mathcal{T}}_K} \min_{v_p \in \mathcal{P}_{p_K}(L)} \|\nabla(v - v_p)\|_L^2 \quad \forall v \in H_{0,D}^1(\Omega), \quad (1.3) $$
(approximation $p$-robustly equivalent to elementwise $H^1$-orthogonal projector)

where $p_K$ is the minimal polynomial degree over $\tilde{\mathcal{T}}_K$, see (2.7). The generic constants hidden in (1.2) and (1.3) only depend on local mesh shape-regularity, local polynomial-variation parameter, and the space dimension $d$, which improves the results in [6, 19, 3, 26, 16, 30, 27] discussed above. With respect to [35, 10], (1.2) and (1.3) are less local (two layers of neighbors in place of one) but the constants hidden in $\lesssim$ are uniform in the polynomial degree $p$. Details are given in Definition 3.3 and Theorem 3.4. The properties of $P_{hp}$ immediately lead to two important consequences. Let $v \in H_{0,D}^1(\Omega)$. First, for a uniform polynomial degree $p \ge 1$ for simplicity,

$$ \min_{v_{hp} \in \mathcal{P}_p(\mathcal{T}_h) \cap H_{0,D}^1(\Omega)} \|\nabla(v - v_{hp})\|^2 \approx \sum_{K \in \bar{\mathcal{T}}_h} \min_{v_p \in \mathcal{P}_p(K)} \|\nabla(v - v_p)\|_K^2, \quad (1.4) $$
($p$-robust global continuous – local discontinuous equivalence)

where the hidden constant only depends on mesh shape regularity and $d$. The full version of this result, also considering variable polynomial degree, is stated in Theorem 3.5. This improves [35, Theorem 2] and [10, Theorem 4], where the equivalence constant was unfavorably dependent on the polynomial degree $p$.
Second, (1.3) immediately implies the optimal **hp** approximation bound

$$ \|\nabla(v - P_{hp}v)\|_K^2 \lesssim \sum_{L \in \mathcal{T}_K} \left( \frac{h_L^{\min(s_L-1, \underline{p}_K)}}{\underline{p}_K^{s_L-1}} \|v\|_{H^{s_L}(L)} \right)^2 \quad \forall K \in \mathcal{T}_h \quad (1.5) $$

/optimal elementwise hp approximation estimate)

whenever the approximated function $v \in H_{0,D}^1(\Omega)$ additionally has, separately on each mesh element $L \in \mathcal{T}_h$, the Sobolev regularity

$$ v|_L \in H^{s_L}(L) \text{ for a Sobolev regularity exponent } s_L \ge 1. \quad (1.6) $$

The constant hidden in $\lesssim$ in (1.5) only depends on local mesh shape-regularity, local polynomial-variation parameter, space dimension $d$, and the regularity exponents $s_L$. This extends [35, equation (3)] to the **hp**-setting while removing the possible dependence of the generic constant on the polynomial degree **p** and [27, Corollary 3.5] to approximated functions $v$ with the regularity only requested elementwise ($v|_\omega \in H^s(\omega)$ for some patch subdomains $\omega \subset \Omega$ bigger than the individual mesh elements and $s > 1$ are not needed). Details form the content of Theorem 3.6.

## 1.4 Crucial tools: polynomial extension operators and p-stable decompositions

There are two crucial tools used to obtain the above results. First, these are polynomial extension operators. After the seminal contributions in Gagliardo [25] and Muñoz-Sola [31], these have been obtained in Ainsworth and Demkowicz [2] in two space dimensions and in Demkowicz et al. [17] in three space dimensions, see also the references therein. We more precisely employ their broken extensions on patches of elements, obtained in Ern and Vohralík [23, Theorems 2.2 and 2.4, Corollaries 3.1 and 3.7], following Braess et al. [8] (cf. also Chaumont-Frelet and Vohralík [13, Corollaries 3.4 and 4.4]). We actually need to generalize these results further to larger (extended) patches and no trace boundary conditions. Second, these are **p**-robust stable decompositions. We will namely use that of Schöberl et al. [32] for a uniform polynomial degree in both two and three space dimensions and that of Karkulik et al. [28] for a variable polynomial degree in two space dimensions

## 1.5 Organization of this manuscript

We set up the notation in Section 2. We then present our main results in full details in Section 3. The more involved proofs are collected in Sections 4 and 5. We also state four independent results in the Appendix. We first formulate the **p**-stable decomposition results from [32, 28] in a form suitable for us in Appendix A. We then study patch enumerations/shellability of simplicial complexes in Appendix B. We finally generalize the results from [23, 13] to larger (extended) patches and no trace boundary conditions in Appendix C.

This contribution only concerns the $H^1$ case. Extensions to the $H(\operatorname{div})$ context are addressed in Demkowicz and Vohralík [18].

# 2 Setting and notation

In this section, we collect the notation.

## 2.1 Simplicial mesh $\mathcal{T}_h$

Let $\mathcal{T}_h$ be a simplicial mesh of $\Omega$, i.e., a finite collection of nontrivial closed triangles or tetrahedra $K$ covering $\bar{\Omega}$, where the intersection of two different simplices is either empty or their common vertex, edge, or face. The element and mesh shape-regularity parameters $\kappa_K$ and $\kappa_h$ are respectively given by

$$ \kappa_K := \frac{h_K}{\rho_K}, \qquad \kappa_h := \max_{K \in \mathcal{T}_h} \kappa_K, \quad (2.1) $$

where $h_K$ is the diameter of the simplex $K$ and $\rho_K$ that of the largest ball contained in $K$. The vector **h** collects the element diameters, $h = \{h_K\}_{K \in \mathcal{T}_h}$. Uniformly bounded $\kappa_h$ allows for families of strongly graded meshes with local refinements ($h$ highly nonuniform), but not for anisotropic elements.
<figure><img src="image_2.png" /><figcaption>Figure 1: Vertex patch $\mathcal{T}_a$ for $a \in V_h$ in the interior of $\Omega$ (left) and on the boundary of $\Omega$ (right)</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Figure 2: Extended vertex patch $\tilde{\mathcal{T}}_a$ for $a \in V_h$ in the interior of $\Omega$ (left) and extended element patch $\tilde{\mathcal{T}}_K$ for $K \in \mathcal{T}_h$ touching $\partial\Omega$ (right) (both generated by the marked vertex patches $\mathcal{T}_b$)</figcaption></figure>

## 2.2 Vertices, edges, faces, and patches of mesh elements

For a simplex $K \in \mathcal{T}_h$, denote by $\mathcal{F}_K$ the set of its $(d-1)$-dimensional faces and by $V_K$ the set of its vertices. Let $V_h$ collect all mesh vertices. For a vertex $a \in V_h$, denote by $\mathcal{T}_a$ the patch of the elements of $\mathcal{T}_h$ that share $a$ and by $\omega_a$ the corresponding subdomain, see Figure 1. We will also need the extended vertex patch $\tilde{\mathcal{T}}_a$ and the corresponding subdomain $\tilde{\omega}_a$; this is given by all elements from $\mathcal{T}_b$, $b$ a vertex of $\mathcal{T}_a$, see Figure 2 (left) for an illustration. For a simplex $K \in \mathcal{T}_h$, let $\tilde{\mathcal{T}}_K$ be the extended element patch given by the extended vertex patches $\tilde{\mathcal{T}}_a$ for all vertices of $a$ of $K$, see Figure 2 (right) for an illustration. $\tilde{\mathcal{T}}_K$ comprises $K$ and all $L$ sharing a vertex with $K$ or its vertex neighbor; the corresponding subdomain is $\tilde{\omega}_K$. We collect the vertices from respectively $\tilde{\mathcal{T}}_a$ and $\tilde{\mathcal{T}}_K$ in the sets $\tilde{V}_a$ and $\tilde{V}_K$. Diameters of respectively $\omega_a$, $\tilde{\omega}_a$, and $\tilde{\omega}_K$ are denoted by $h_{\omega_a}$, $h_{\tilde{\omega}_a}$, and $h_{\tilde{\omega}_K}$.

## 2.3 Hat functions and the partition of unity

Let the continuous, piecewise first-order polynomial (affine) “hat” function $\psi^a$ take value 1 at the vertex $a$ and zero at all the other vertices. We note that $\omega_a$ corresponds to the support of $\psi^a$, and that these functions form the partition of unity

$$ \sum_{a \in V_h} \psi^a = 1. \qquad (2.2) $$

## 2.4 Boundary subsets $\Gamma_D$ and $\Gamma_N$

Let $\Gamma_D$, $\Gamma_N$ be two disjoint and relatively open, possibly empty subsets of the boundary $\partial\Omega$ such that $\partial\Omega = \overline{\Gamma_D} \cup \overline{\Gamma_N}$. We also require that $\Gamma_D$ and $\Gamma_N$ have polygonal Lipschitz boundaries and we assume that each boundary face of the mesh $\mathcal{T}_h$ lies entirely either in $\overline{\Gamma_D}$ or in $\overline{\Gamma_N}$.
## 2.5 The spaces $H^1$ on the entire computational domain and its subdomains

Let $\omega \subseteq \Omega$ correspond to a patch. We let $L^2(\omega)$ be the space of scalar-valued square-integrable functions defined on $\omega$ and we denote by $L^2(\omega) := [L^2(\omega)]^d$ vector-valued functions with each component in $L^2(\omega)$. Let $\|\cdot\|_{\omega}$ stand for the $L^2(\omega)$ or $L^2(\omega)$ norm and $(\cdot, \cdot)_{\omega}$ for the corresponding scalar product; we drop the index when $\omega = \Omega$. Then, $H^1(\omega) = \{v \in L^2(\omega); \nabla v \in L^2(\omega)\}$. Moreover, if $\partial\omega \cap \Gamma_D$ is of nonzero measure ($\partial\omega$ contains at least one face from the Dirichlet boundary $\bar{\Gamma}_D$), then we use

$$H_{0,D}^{1}(\omega) := \{v \in H^{1}(\omega); v = 0 \text{ on } (\partial\omega \cap \Gamma_{D})^{\circ}\}. \qquad (2.3)$$

Let $a \in \mathcal{V}_h$ be a mesh vertex. For the vertex patch subdomain $\omega_a$, cf. Figure 1, we will employ the specific notation $H_{0,D,\psi^a}^1(\omega_a)$ for the subspace of $H^1(\omega_a)$ with zero trace on those faces in $\partial\omega_a$ where the hat function $\psi^a$ vanishes (all $\partial\omega_a$ for interior vertices) or which lie on the Dirichlet boundary $\bar{\Gamma}_D$,

$$H_{0,D,\psi^a}^{1}(\omega_a) := \{v \in H^1(\omega_a); v = 0 \text{ on } \partial\omega_a \cap \{\psi^a = 0\} \text{ and on } (\partial\omega_a \cap \Gamma_D)^{\circ}\}. \qquad (2.4)$$

In Figure 1, this respectively corresponds to the double line (for interior patches $\mathcal{T}_a$, left) or to the double and zigzag lines (for boundary patches $\mathcal{T}_a$, right). Congruently, for an arbitrary patch subdomain $\omega$ and a vertex $a$ therein, we let

$$H_{0,D,\psi^a}^{1}(\omega_a, \omega) := \{v \in H^1(\omega_a \cap \omega); v = 0 \\ \text{on } \partial(\omega_a \cap \omega) \cap \{\psi^a = 0\} \text{ and on } (\partial(\omega_a \cap \omega) \cap \Gamma_D)^{\circ}\}. \qquad (2.5)$$

This is as above in (2.4) but not for the vertices $a$ on the boundary of $\omega$: functions from $H_{0,D,\psi^a}^1(\omega_a, \omega)$ do not vanish on $\partial(\omega_a \cap \omega)$ unless this is a part of $\bar{\Gamma}_D$.

## 2.6 Discontinuous piecewise polynomials

We suppose that with each mesh element $K \in \mathcal{T}_h$, there is an associated polynomial degree $p_K \ge 1$. We collect these elementwise polynomial degrees into the vector $p = \{p_K\}_{K \in \mathcal{T}_h}$. This is a general variable polynomial degree case. In turn, we call a uniform polynomial degree case the situation where $p_K = p \ge 1$ for all $K \in \mathcal{T}_h$. For each simplex $K \in \mathcal{T}_h$, we denote by $\mathcal{P}_{p_K}(K)$ the space of scalar-valued polynomials on $K$ of total degree at most $p_K$. We then let $\mathcal{P}_p(\mathcal{T}_h)$ stand for discontinuous piecewise polynomials,

$$\mathcal{P}_p(\mathcal{T}_h) := \{v_h \in L^2(\Omega); v_h|_K \in \mathcal{P}_{p_K}(K) \quad \forall K \in \mathcal{T}_h\}, \qquad (2.6)$$

with $\mathcal{P}_p(\mathcal{T}_h)$ highlighting the uniform polynomial degree case. To form the usual continuous (conforming) finite element spaces, we will write $\mathcal{P}_p(\mathcal{T}_h) \cap H^1(\Omega)$ (this is equivalent to $\mathcal{P}_p(\mathcal{T}_h) \cap C^0(\bar{\Omega})$) and similarly for the subspaces reflecting the boundary conditions. The same notation will also be used on various patches. For $v_{hp} \in \mathcal{P}_p(\mathcal{T}_h)$, $\nabla v_{hp}$ stands for the broken (elementwise) gradient, $(\nabla v_{hp})|_K := \nabla(v_{hp})|_K$, $K \in \mathcal{T}_h$.

For $K \in \mathcal{T}_h$, the smallest polynomial degree over the extended element patch $\tilde{\mathcal{T}}_K$,

$$\underline{p}_K := \min_{L \in \mathcal{T}_K} \{p_L\}, \qquad (2.7)$$

leads to local and global polynomial-variation parameters $\kappa_{p_K}$ and $\kappa_p$

$$\kappa_{p_K} := \frac{p_K}{\underline{p}_K}, \qquad \kappa_p := \max_{K \in \mathcal{T}_h} \kappa_{p_K}. \qquad (2.8)$$

Note that $\kappa_p = 1$ if the polynomial degree $p$ is uniform.

## 2.7 Discontinuous piecewise polynomials with degree lowered according to the neighbors

Let $K \in \mathcal{T}_h$. Let simply $\underline{\mathcal{P}}_{p_K}(K) := \mathcal{P}_{p_K}(K) = \mathcal{P}_p(K)$ if the polynomial degree $p$ is uniform. In general, we let $\underline{\mathcal{P}}_{p_K}(K)$ be the subspace of $\mathcal{P}_{p_K}(K)$ where the restriction to each face $F_L \in \mathcal{F}_K$ between the element $K$ and its face neighbor $L \in \mathcal{T}_h$ is a polynomial of degree $\min\{p_K, p_L\}$ and, if $d = 3$, additionally the restriction to each edge $e$ of $K$ is a polynomial of degree $\min_{L \in \mathcal{T}_e}\{p_L\}$, where $\mathcal{T}_e$ is the set of all
tetrahedra having e as edge. The space $\mathcal{P}_{p_K}(K)$ is “aware” of the polynomial degrees of the edge and face neighbors of K in $\mathcal{T}_h$. We then define

$$ \underline{\mathcal{P}_p(\mathcal{T}_h)} := \{v_h \in L^2(\Omega); v_h|_K \in \underline{\mathcal{P}_{p_K}(K)} \quad \forall K \in \mathcal{T}_h\}. \quad (2.9) $$

Similar notation is used on submeshes. In the uniform polynomial degree case, $\underline{\mathcal{P}_p(\mathcal{T}_h)} = \mathcal{P}_p(\mathcal{T}_h)$. For a general variable polynomial degree, there holds

$$ \underline{\mathcal{P}_p(\mathcal{T}_h)} \cap H_{0,D}^1(\Omega) = \mathcal{P}_p(\mathcal{T}_h) \cap H_{0,D}^1(\Omega), \quad (2.10) $$

since the trace-continuity requirement in $H_{0,D}^1(\Omega)$ exactly leads to the polynomial order decrease of $\underline{\mathcal{P}_p(\mathcal{T}_h)}$ with respect to $\mathcal{P}_p(\mathcal{T}_h)$. We, however, note that in general

$$ \underline{\mathcal{P}_p(\mathcal{T}_\omega)} \cap H^1(\omega) \subset \mathcal{P}_p(\mathcal{T}_\omega) \cap H^1(\omega), \quad \underline{\mathcal{P}_p(\mathcal{T}_\omega)} \cap H_{0,D}^1(\omega) \subset \mathcal{P}_p(\mathcal{T}_\omega) \cap H_{0,D}^1(\omega) \quad (2.11) $$

for a submesh $\mathcal{T}_\omega$ of $\mathcal{T}_h$ and the corresponding $\omega \subsetneq \Omega$. Indeed, the polynomial order decrease in $\underline{\mathcal{P}_p(\mathcal{T}_\omega)}$ also applies on the boundary of $\omega$ on the above left-hand sides.

## 2.8 Elementwise Lagrange interpolation operator

Following Ciarlet [14, Section 2.2] or Ern and Guermond [20, Section 7.4], we let

$$ I_{hp}^{\mathcal{L}} : \{v \in L^2(\Omega); v|_K \in C^0(K) \quad \forall K \in \mathcal{T}_h\} \to \underline{\mathcal{P}_p(\mathcal{T}_h)} $$

be prescribed, separately on each $K \in \mathcal{T}_h$, by

$$ I_{hp}^{\mathcal{L}}(v)|_K(x) = (v|_K)(x) \quad \text{in all Lagrange nodes } x \text{ of the space } \underline{\mathcal{P}_{p_K}(K)}. \quad (2.12) $$

We will only employ $I_{hp}^{\mathcal{L}}$ on discontinuous piecewise polynomials; recall that $I_{hp}^{\mathcal{L}}$ cannot be applied to functions $v \in H_{0,D}^1(\Omega)$ which may not admit point values.

## 2.9 Notation $\lesssim$

We will use the notation $a \lesssim b$ when there holds $a \le Cb$ for a positive constant $C$ and $a \approx b$ when $a \lesssim b$ and $b \lesssim a$ hold simultaneously. All dependencies of $C$ will systematically be given. In any case, all such constants $C$ in this manuscript are independent of the mesh size $h$ and of the polynomial degree $p$.

# 3 Main results

We give here our main results. We rely on two assumptions:

**Assumption 3.1 (Patch subdomains).** For a mesh element $K \in \mathcal{T}_h$, let $\hat{\mathcal{V}}_K$ be a subset of vertices $\mathcal{V}_K$ and $\hat{\mathcal{T}}_K := \cup_{a \in \hat{\mathcal{V}}_K} \{\hat{T}_a\}$ an extended patch as per Section 2.2. We suppose that the associated open subdomain $\hat{\omega}_K$ is Lipschitz, its closure is contractible, and its boundary does not coincide with the Dirichlet boundary, $\partial\hat{\omega}_K \neq \Gamma_D$.

**Assumption 3.2 (Variable polynomial degree $p$-stable $H^1$ decomposition in 3D).** Suppose that [28, Theorem 2.5] holds for $d=3$ and variable polynomial degree $p$.

Assumption 3.1 is similar to those in [24, 3]. Remark that it does not request $\bar{\Omega}$ to be contractible but merely the patch subdomains. It may not be satisfied for a coarse triangulation, but typically will be satisfied for a finer mesh.

Assumption 3.2 only concerns the case of variable polynomial degree in three space dimensions. For a uniform polynomial degree, the necessary stable decomposition is available from Schöberl et al. [32] for both $d=2,3$. For the variable polynomial degree, Karkulik et al. [28, Theorem 2.5] give it for $d=2$. We believe that it is satisfied for $d=3$ as well, but a rigorous proof is outside of the scope of this paper.
## 3.1 A p-stable local projector in $H^1$

We proceed in five steps: 1) elementwise $H^1(K)$-orthogonal projection (local-best approximation) (a mean value is fixed for uniqueness), giving a discontinuous piecewise polynomial; 2) patchwise potential reconstruction yielding a continuous piecewise polynomial; this stage employs the hat functions $\psi^a$ from (2.2) and the elementwise Lagrange projector $I_{hp}^L$ from (2.12) to gain locality and projection (not p-robust); 3) correction by element bubbles for a suitable elementwise mean value; 4) patchwise potential reconstruction of the remainder (by broken polynomial extension of Appendix C) and p-stable decomposition of Appendix A ($\psi^a$ and $I_{hp}^L$ are not used); 5) combination of the previous steps.

### 3.1.1 Definition of the projector

Recall the notation of Section 2.

**Definition 3.3** (A p-stable local projector in $H^1$). We define

$$P_{hp}: H_{0,D}^{1}(\Omega) \to \mathcal{P}_{p}(\mathcal{T}_{h}) \cap H_{0,D}^{1}(\Omega) \quad (3.1)$$

as follows. Let $v \in H_{0,D}^1(\Omega)$.

**I.** Define the discontinuous piecewise polynomial $\tau_{hp}$ by the elementwise $H^1(K)$-orthogonal projection of $v$ onto $\mathcal{P}_{p_K}(K)$ (without any trace prescription)

$$\tau_{hp} \in \mathcal{P}_{p}(\mathcal{T}_{h}), \quad \tau_{hp}|_{K} := \arg \min_{\substack{v_p \in \mathcal{P}_{p_K}(K) \\ (v_p, 1)_K = (v, 1)_K}} \|\nabla(v - v_p)\|_K. \quad (3.2)$$

(elementwise projection)

**II.** Starting from $\tau_{hp}$:

(a) On each vertex patch $\mathcal{T}_a$, $a \in V_h$, cf. Figure 3 (left), define the continuous piecewise polynomial $s_p^a \in \mathcal{P}_p(\mathcal{T}_a) \cap H_{0,D,\psi^a}(\omega_a)$ via

$$s_p^a := \arg \min_{v_p \in \mathcal{P}_p(\mathcal{T}_a) \cap H_{0,D,\psi^a}^1(\omega_a)} \|\nabla(I_{hp}^L(\psi^a \tau_{hp}) - v_p)\|_{\psi^a}. \quad (3.3a)$$

(patchwise potential reconstruction)

(b) Extending $s_p^a$ by zero outside of the patch subdomain $\omega_a$, define

$$s_{hp} := \sum_{a \in V_h} s_p^a \in \mathcal{P}_p(\mathcal{T}_h) \cap H_{0,D}^1(\Omega). \quad (3.3b)$$

(potential reconstruction)

**III.** Starting from the provisional remainder $\tau_{hp} - s_{hp}$, let

$$\delta_{hp} \in \mathcal{P}_p(\mathcal{T}_h) \cap H_{0,D}^1(\Omega), \quad \delta_{hp}|_K := \begin{cases} 0 & \text{if } p_K < d+1, \quad (3.4a) \\ \frac{(\tau_{hp} - s_{hp}, 1)_K}{(b_K, 1)_K} b_K & \text{if } p_K \ge d+1, \quad (3.4b) \end{cases}$$

(elementwise mean value bubble correction if $p_K \ge d+1$)

where $b_K$ is the bubble on $K$, $b_K \in \mathcal{P}_{d+1}(K) \cap H_0^1(K)$, $b_K = \Pi_{a \in V_h}(\psi^a)|_K$.

**IV.** Starting from the remainder $\tau_{hp} - s_{hp} - \delta_{hp}$:

(a) On each extended vertex patch $\tilde{\mathcal{T}}_a$, $a \in V_h$, cf. Figure 3 (right), define

$$\zeta_p^a := 0 \quad \text{if } p_K < d+1 \text{ for at least one } K \in \tilde{\mathcal{T}}_a, \quad (3.5a)$$

or the continuous piecewise polynomial $\zeta_p^a \in \mathcal{P}_p(\tilde{\mathcal{T}}_a) \cap H_{0,D}^1(\tilde{\omega}_a)$ via

$$\zeta_p^a := \arg \min_{\substack{v_p \in \mathcal{P}_p(\tilde{\mathcal{T}}_a) \cap H_{0,D}^1(\tilde{\omega}_a) \\ (v_p, 1)_{\tilde{\omega}_a} = (\tau_{hp} - s_{hp} - \delta_{hp}, 1)_{\tilde{\omega}_a} = 0 \text{ if } (\partial \tilde{\omega}_a \cap \Gamma_D)^\circ = \emptyset}} \|\nabla(\tau_{hp} - s_{hp} - \delta_{hp} - v_p)\|_{\tilde{\omega}_a} \quad (3.5b)$$

(patchwise remainder potential reconstruction)
<figure><img src="image_4.png" /><figcaption>Figure 3: The standard non-$\mathbf{p}$-robust potential reconstruction component $s_p^a$ from (3.3a) (left) and the $\mathbf{p}$-robust correction $\zeta_p^a$ from (3.5b) together with its $\mathbf{p}$-stable decomposition (3.6b); only the “interior” component $\zeta_p^{a,a}$ is used (right)</figcaption></figure>

(b) On each extended vertex patch $\tilde{T}_a$, $a \in V_h$, cf. Figure 3 (right), set

$$ \zeta_p^{a,a} := 0 \quad \text{if } p_K < d+1 \text{ for at least one } K \in \tilde{T}_a, \qquad (3.6a) $$

or apply to $\zeta_p^a$ from (3.5b) the $\mathbf{p}$-stable decomposition of Corollary A.1,

$$ \zeta_p^a = \sum_{b \in \tilde{V}_a} \zeta_p^{a,b} \quad \text{with} \quad \zeta_p^{a,a} \in \mathcal{P}_p(T_a) \cap H_{0,D}^1(\Omega_a) \\ \text{if } p_K \ge d+1 \text{ for all } K \in \tilde{T}_a. \qquad (3.6b) $$

(patchwise $\mathbf{p}$-stable reconstructed remainder decomposition)

(c) Extending the “interior” component $\zeta_p^{a,a}$ by zero outside of $\omega_a$, define

$$ \zeta_{hp} := \sum_{a \in V_h} \zeta_p^{a,a} \in \mathcal{P}_p(T_h) \cap H_{0,D}^1(\Omega). \qquad (3.7) $$

(p-robust correction $\zeta_{hp}$ without employment of $\psi^a$ and $I_{hp}^L$)

V. Define the final continuous piecewise polynomial

$$ P_{hp}(v) := s_{hp} + \delta_{hp} + \zeta_{hp} \in \mathcal{P}_p(T_h) \cap H_{0,D}^1(\Omega). \qquad (3.8) $$

We will justify all the steps of Definition 3.3 in Section 4 below.

3.1.2 Design principles

Let us stress the design principles of Definition 3.3.

1. The construction of $\tau_{hp}$ in step I. sets our local-best discontinuous projection “target”. There holds $\tau_{hp} \in \mathcal{P}_p(T_h)$ but in general $\tau_{hp} \notin H_{0,D}^1(\Omega)$. In the rest of Definition 3.3, we search to stay in $\mathcal{P}_p(T_h)$, as close as possible to $\tau_{hp}$, and keeping its approximation power, but recovering $H_{0,D}^1(\Omega)$-conformity.

2. The construction of $s_{hp}$ in step II. is identical to [22, Construction 3.8 and Remark 3.10] and [23, Corollaries 3.1 and 3.7 together with equation (4.3)] from a posteriori error analysis. This is not $\mathbf{p}$-robust since the cut-off by the hat functions $\psi^a$ from (2.2) increases the polynomial degree by one and is brought back down to $\mathbf{p}$ by the canonical elementwise projector $I_{hp}^L$ from (2.12). We gain here the projection property: if $v \in \mathcal{P}_p(T_h) \cap H_{0,D}^1(\Omega)$, then $\tau_{hp} = s_{hp} = v$ and the “provisional remainder” $\tau_{hp} - s_{hp}$ vanishes (then the remaining steps give $\delta_{hp} = \zeta_{hp} = 0$).

3. The purpose of the construction of $\delta_{hp}$ in step III. is to obtain the (final) remainder $\tau_{hp} - s_{hp} - \delta_{hp}$ with vanishing mean values on all mesh elements $K \in T_h$ with the polynomial degree at least equal to $d+1$ (employing bubbles as in [21, equation (3.16)]). Note that $\delta_{hp}|_K$, if nonzero, is only a $(d+1)$-degree polynomial.
4. The construction of $\zeta_{hp}$ in step IV. is the salient feature for **p**-robustness. Neither the hat functions $\psi^a$ nor the elementwise projector $I_{hp}^c$ are used at this stage. First, in (3.5b), we employ a potential reconstruction similar to (3.3a) which however does not impose zero trace on $\partial\tilde{\omega}_a$ (except for $\partial\tilde{\omega}_a \cap \Gamma_D$). Second, in (3.6b), **p**-stable decomposition following [32] or [28] is applied (this cannot be applied directly to the remainder $\tau_{hp} - s_{hp} - \delta_{hp}$ which is a discontinuous piecewise polynomial. The zero mean values prepared above are crucial to remove global low order component in the decomposition. In (3.7), we then merely employ the “interior” component.

5. In step V., $P_{hp}(v)$ is defined as $s_{hp}$ corrected by $\delta_{hp} + \zeta_{hp}$.

### 3.1.3 Properties of the projector

Recall the minimal polynomial degree $p_K$ from (2.7). The following theorem summarizes the properties of the projector $P_{hp}$, improving on [34, 7, 16, 30, 24, 27, 6, 19, 3, 26]. The proof is given in Section 5.

**Theorem 3.4** (Projection, stability, and approximation of $P_{hp}$). Let a simplicial mesh $\mathcal{T}_h$ of $\Omega$ and a polynomial degree **p** be given. Let Assumptions 3.1 and 3.2 hold. The operator $P_{hp} : H_{0,D}^1(\Omega) \to \mathcal{P}_p(\mathcal{T}_h) \cap H_{0,D}^1(\Omega)$ from Definition 3.3 satisfies

$$ P_{hp}(v) = v \quad \forall v \in \mathcal{P}_p(\mathcal{T}_h) \cap H_{0,D}^1(\Omega). \qquad (3.9) \\ \text{(projection)} $$

Moreover, for any function $v \in H_{0,D}^1(\Omega)$ and any mesh element $K \in \mathcal{T}_h$, there holds

$$ \|\nabla P_{hp}(v)\|_K^2 \lesssim \sum_{L \in \mathcal{T}_K} \|\nabla v\|_L^2, \qquad (3.10) \\ \text{(local H}^1\text{-seminorm stability)} $$

$$ \|\nabla(v - P_{hp}(v))\|_K^2 \lesssim \sum_{L \in \mathcal{T}_K} \min_{v_p \in \mathcal{P}_{\mathcal{P}_K}(L)} \|\nabla(v - v_p)\|_L^2. \qquad (3.11) \\ \text{(approximation p-robustly equivalent to elementwise H}^1\text{-orthogonal projector)} $$

The constant hidden in $\lesssim$ only depends on the local mesh shape-regularity $\max_{L \in \mathcal{T}_K} \kappa_L$ with $\kappa_L$ given by (2.1), the local polynomial-variation parameter $\max_{L \in \mathcal{T}_K} \kappa_{pL}$ with $\kappa_{pL}$ given by (2.8), and the space dimension $d$.

## 3.2 **p**-robust equivalence of continuous and discontinuous approximation in $H^1$

The following result improves [35, Theorem 2] and [10, Theorem 4], removing the dependence of the equivalence constant on the polynomial degree **p**.

**Theorem 3.5** (**p**-robust equivalence of global-best and local-best approximations in $H^1$). Let $v \in H_{0,D}^1(\Omega)$, a simplicial mesh $\mathcal{T}_h$ of $\Omega$, and a polynomial degree **p** be given. Let Assumptions 3.1 and 3.2 hold. Then

$$ \min_{v_{hp} \in \mathcal{P}_p(\mathcal{T}_h) \cap H_{0,D}^1(\Omega)} \|\nabla(v - v_{hp})\|^2 \approx \sum_{K \in \mathcal{T}_h} \min_{v_p \in \mathcal{P}_p(K)} \|\nabla(v - v_p)\|^2_K \qquad (3.12a) \\ \text{(p-robust global continuous – local discontinuous equivalence)} $$

if the polynomial degree is uniform, $p_K = p \ge 1$ for all $K \in \mathcal{T}_h$, and

$$ \min_{v_{hp} \in \mathcal{P}_p(\mathcal{T}_h) \cap H_{0,D}^1(\Omega)} \|\nabla(v - v_{hp})\|^2 \lesssim \sum_{K \in \mathcal{T}_h} \sum_{L \in \mathcal{T}_K} \min_{v_p \in \mathcal{P}_{\mathcal{P}_K}(L)} \|\nabla(v - v_p)\|^2_L, \qquad (3.12b) $$

$$ \min_{v_{hp} \in \mathcal{P}_p(\mathcal{T}_h) \cap H_{0,D}^1(\Omega)} \|\nabla(v - v_{hp})\|^2 \gtrsim \sum_{K \in \mathcal{T}_h} \sum_{L \in \mathcal{T}_K} \min_{v_p \in \mathcal{P}_{\mathcal{P}_K}(L)} \|\nabla(v - v_p)\|^2_L \qquad (3.12c) $$

if the polynomial degree is variable, $p_K \ge 1$ for all $K \in \mathcal{T}_h$, $p_K \ne p$. The hidden constant only depends on the mesh shape-regularity parameter $\kappa_h$ given by (2.1), the polynomial-variation parameter $\kappa_p$ given by (2.8), and the space dimension $d$.
Proof. As for (3.12a), for uniform $p$, $\mathcal{P}_p(\mathcal{T}_h) \cap H_{0,D}^1(\Omega) \subset \mathcal{P}_p(\mathcal{T}_h)$, so that the inequality $\gtrsim$ (actually $\ge$) follows trivially. For the $\lesssim$ inequality, employing the projector $P_{hp}$ from Definition 3.3, using (3.11) from Theorem 3.4, and reasoning by a finite overlap thanks to the mesh shape regularity yield, we have

$$ \begin{align*} \min_{v_{hp} \in \mathcal{P}_p(\mathcal{T}_h) \cap H_{0,D}^1(\Omega)} \|\nabla(v - v_{hp})\|^2 &\stackrel{(3.1)}{\le} \|\nabla(v - P_{hp}(v))\|^2 = \sum_{K \in \mathcal{T}_h} \|\nabla(v - P_{hp}(v))\|_K^2 \\ &\lesssim \sum_{K \in \mathcal{T}_h} \sum_{L \in \tilde{\mathcal{T}}_K} \min_{v_p \in \mathcal{P}_p(L)} \|\nabla(v - v_p)\|_L^2 \lesssim \sum_{K \in \mathcal{T}_h} \min_{v_p \in \mathcal{P}_p(K)} \|\nabla(v - v_p)\|_K^2. \end{align*} $$

As for (3.12b), for variable $p$, the $\lesssim$ inequality is as above, though we are obliged to keep the lowered polynomial degree $\underline{p}_K$ from (2.7) and we cannot combine the double sum into a single one. Congruently, the inequality $\gtrsim$ does not hold in (3.12b), since, recalling (2.11), for $K \in \mathcal{T}_h$, $(\mathcal{P}_p(\mathcal{T}_h) \cap H_{0,D}^1(\Omega))|_{\tilde{\omega}_K}$ may contain functions that are not lowered-degree polynomials from $\mathcal{P}_{\underline{p}_K}(\tilde{\mathcal{T}}_K)$. Inequality (3.12c) rectifies this. $\square$

### 3.3 Optimal local hp approximation estimates under minimal elementwise Sobolev regularity in $H^1$

We finally focus on functions with additional $H^{sL}(L)$ regularity, $s_L \ge 1$, only requested locally on each mesh element $L \in \mathcal{T}_h$. The following theorem extends [35, equation (3)] to the hp-setting while removing the possible (unfavorable) dependence of the generic constant on the polynomial degree $p$. In comparison with [27, Corollary 3.5], the regularity of the approximated function $v$ is only requested elementwise and not patchwise, but the dependency region contains an additional layer of neighbors. We note that the unknown constant hidden in $\lesssim$ in (3.14), though independent of the polynomial degree $p$, depends on the local regularity exponents $s_L$ in a not explicit way, so that (3.14) does not automatically yield exponential convergence in terms of the polynomial degree for analytic functions.

**Theorem 3.6 (hp-optimal approximation estimate in $H^1$ under minimal elementwise Sobolev regularity).** Let a simplicial mesh $\mathcal{T}_h$ of $\Omega$ and a polynomial degree $p$ be given. Let Assumptions 3.1 and 3.2 hold. Let, separately on each $L \in \mathcal{T}_h$,

$$ v|_L \in H^{sL}(L) \text{ for a Sobolev regularity exponent } s_L \ge 1. \quad (3.13) $$

Then the projector $P_{hp}$ of Definition 3.3 satisfies

$$ \|\nabla(v - P_{hp}v)\|_K^2 \lesssim \sum_{L \in \tilde{\mathcal{T}}_K} \left( \frac{h_L^{\min(s_L-1, \underline{p}_K)}}{\underline{p}_K^{s_L-1}} \|v\|_{H^{sL}(L)} \right)^2 \quad \forall K \in \mathcal{T}_h. \quad (3.14) $$

(optimal elementwise hp approximation estimate)

The constant hidden in $\lesssim$ only depends on the local mesh shape-regularity $\max_{L \in \tilde{\mathcal{T}}_K} \kappa_L$ with $\kappa_L$ given by (2.1), the local polynomial-variation parameter $\max_{L \in \tilde{\mathcal{T}}_K} \kappa_{pL}$ with $\kappa_{pL}$ given by (2.8), the space dimension $d$, and the regularity exponents $s_L$.

**Proof.** Let $K \in \mathcal{T}_h$. The claim (3.14) follows from approximation property (3.11) and from well-known hp-approximation bounds, see, e.g., [5, Lemma 4.1]. $\square$

**Remark 3.7 (L$^2$ approximation estimate).** Let the assumptions of Theorem 3.6 be satisfied. Then, with the same notation,

$$ \|v - P_{hp}v\|_K^2 \lesssim \sum_{L \in \tilde{\mathcal{T}}_K} \left( \frac{h_L^{\min(s_L, \underline{p}_K) + 1}}{\underline{p}_K^{s_L-1}} \|v\|_{H^{sL}(L)} \right)^2 \quad \forall K \in \mathcal{T}_h. \quad (3.15) $$

Indeed, proceeding as in (5.11) below and noting that $v - \tau_{hp}$ and $\tau_{hp} - s_{hp} - \delta_{hp} - \tilde{\zeta}_p^K$ are of mean value 0 on the element $K$ by respectively (3.2) and (5.4) and that $\zeta_p^{K,a,a} - \zeta_p^{a,a}$ belong to the homogeneous Dirichlet
space $H_{0,D,\psi^a}^1(\omega_a)$ from (2.4) on the patch $\omega_a$, the Poincaré inequality shows that $h_K^{-1}\|v - P_{hp}(v)\|_K$ is bounded by

$$ \|\nabla(v - \tau_{hp})\|_{\tilde{\omega}_K} + \|\nabla(\tau_{hp} - s_{hp} - \delta_{hp} - \zeta_p^K)\|_{\tilde{\omega}_K} + \sum_{a \in \mathcal{V}_K} \|\nabla(\tilde{\zeta}_p^{K,a,a} - \zeta_p^{a,a})\|_{\omega_a}, $$

in addition to (5.11). Thus, (3.15) follows from (3.14). This $L^2$ approximation estimate is optimal in the mesh size $h$ but suboptimal by $1/p$ in the polynomial degree $p$. Also, we require at least $s_L \ge 1$ Sobolev regularity. $L^2$ projector with optimal hp estimates can be obtained as in Definition 3.3, while working with the $L^2$ norm in place of the $H^1$ norm, in particular starting with $\tau_{hp}|_K := \arg \min_{v_p \in \mathcal{P}_{pK}(K)} \|v - v_p\|_K$ in place of (3.2). Such projector, however, does not seem to lead to (3.11) and (3.12).

# 4 Definition 3.3 of the projector $P_{hp}$

We justify here all steps of Definition 3.3 and summarize the properties of the intermediate objects.

## 4.1 Step I. (elementwise $H^1$ projection $\tau_{hp}$)

$\tau_{hp}$ is clearly well defined by (3.2). There is no interelement continuity requirement, $\tau_{hp} \in \mathcal{P}_p(\mathcal{T}_h)$ is a discontinuous piecewise polynomial, typically not lying in $H_{0,D}^1(\Omega)$.

## 4.2 Step II. (standard potential reconstruction $s_{hp}$)

We have:

**Lemma 4.1** ($s_{hp}$). There holds $s_{hp} \in \mathcal{P}_p(\mathcal{T}_h) \cap H_{0,D}^1(\Omega)$ and $\tau_{hp} - s_{hp} \in \mathcal{P}_p(\mathcal{T}_h)$.

*Proof.* In view of the definition (2.4), all $s_p^a$ from (3.3a) have zero trace on those faces in $\partial\omega_a$ where the hat function $\psi^a$ vanishes or which lie on the Dirichlet boundary $\bar{\Gamma}_D$. Thus it follows that all $s_p^a$ extended by zero outside of the patch subdomains $\omega_a$ belong to $\mathcal{P}_p(\mathcal{T}_h) \cap H_{0,D}^1(\Omega) = \mathcal{P}_p(\mathcal{T}_h) \cap H_{0,D}^1(\Omega)$, recalling (2.10). Consequently, $s_{hp} \in \mathcal{P}_p(\mathcal{T}_h) \cap H_{0,D}^1(\Omega)$. The other claim follows since $\tau_{hp} \in \mathcal{P}_p(\mathcal{T}_h)$, also using (2.10). □

## 4.3 Step III. (elementwise bubble correction $\delta_{hp}$)

For $\delta_{hp}$, there holds:

**Lemma 4.2** ($\delta_{hp}$). There holds $\delta_{hp} \in \mathcal{P}_p(\mathcal{T}_h) \cap H_{0,D}^1(\Omega)$, $\tau_{hp} - s_{hp} - \delta_{hp} \in \mathcal{P}_p(\mathcal{T}_h)$,

$$ (\tau_{hp} - s_{hp} - \delta_{hp}, 1)_K = 0 \quad \forall K \in \mathcal{T}_h \text{ such that } p_K \ge d+1. \quad (4.1) $$

*Proof.* $\delta_{hp} \in \mathcal{P}_p(\mathcal{T}_h) \cap H_{0,D}^1(\Omega)$ follows immediately from the definition (3.4), since $\delta_{hp}|_K$ is either zero, or a polynomial of degree $(d+1) \le p_K$; importantly, $\delta_{hp}$ vanishes on all mesh faces. Then $\tau_{hp} - s_{hp} - \delta_{hp} \in \mathcal{P}_p(\mathcal{T}_h)$ is an immediate consequence of Lemma 4.1. As for (4.1), let a mesh element $K \in \mathcal{T}_h$ with $p_K \ge d+1$ be fixed. By construction, we see from (3.4b)

$$ (\delta_{hp}, 1)_K = \frac{(\tau_{hp} - s_{hp}, 1)_K}{(b_K, 1)_K} (b_K, 1)_K = (\tau_{hp} - s_{hp}, 1)_K. $$

□

## 4.4 Step IV. (p-robust correction $\zeta_{hp}$)

We continue with $\zeta_{hp}$:

**Lemma 4.3** ($\zeta_p^a$ and $\zeta_p^{a,a}$). For each mesh vertex $a \in \mathcal{V}_h$ such that $p_K \ge d+1$ for all $K \in \tilde{\mathcal{T}}_a$, the construction (3.5b) and the decomposition (3.6b) are well defined.
**Proof.** We first note that if $p_K \ge d+1$ for all $K \in \tilde{T}_a$, then (4.1) implies $(\tau_{hp} - s_{hp} - \delta_{hp}, 1)_{\tilde{\omega}_a} = 0$. We have directly included this equality in the definition (3.5b), where we ask $\zeta_p^a$ to either take zero values on the boundary faces from $\partial \tilde{\omega}_a$ lying in $\overline{\Gamma_D}$, or $\zeta_p^a$ to have mean value zero. Thus assumption (A.1) below is satisfied with $T_\omega = \tilde{T}_a$ and $\omega = \tilde{\omega}_a$. Consequently, (3.6b) follows immediately from (A.2) (with $V_\omega = \tilde{V}_a$) and (A.4) in Corollary A.1. Note that we only employ the “interior” component $\zeta_p^{a,a}$; this is from (A.3) supported on the vertex patch subdomain $\omega_a \cap \tilde{\omega}_a$ which is simply $\omega_a$ (no patch truncation happens for the “interior” component, cf. Figure 3 (right)). $\square$

**Lemma 4.4** ($\zeta_{hp}$). There holds $\zeta_{hp} \in \mathcal{P}_p(\tilde{T}_h) \cap H_{0,D}^1(\Omega)$.

**Proof.** As in the proof of Lemma 4.1 above, recalling definition (2.4), all $\zeta_p^a$ extended by zero outside of the patch subdomains $\omega_a$ belong to $H_{0,D}^1(\Omega)$. Thus the inclusion $\zeta_{hp} \in \mathcal{P}_p(\tilde{T}_h) \cap H_{0,D}^1(\Omega)$ follows immediately by (3.7) and (2.10). $\square$

## 4.5 Step V. (combining the previous steps)

We finish by $P_{hp}(v)$:

**Lemma 4.5** ($P_{hp}(v)$). There holds $P_{hp}(v) \in \mathcal{P}_p(\tilde{T}_h) \cap H_{0,D}^1(\Omega)$.

**Proof.** This is an direct consequence of definition (3.8) and of the above lemmas. $\square$

# 5 Proof of Theorem 3.4 on properties of the projector $P_{hp}$

Let the assumptions of Theorem 3.4 be satisfied. We prove the three claims separately.

## 5.1 Projection

Projection is ensured by Steps I.-II. of Definition 3.3:

**Lemma 5.1** (Projection property (3.9)). The projection property (3.9) holds true.

**Proof.** Let $v \in \mathcal{P}_p(\tilde{T}_h) \cap H_{0,D}^1(\Omega)$. Then clearly $\tau_{hp}$ from (3.2) satisfies $\tau_{hp} = v$. Moreover, $I_{hp}^L(\psi^a \tau_{hp}) \in \mathcal{P}_p(\tilde{T}_a) \cap H_{0,D,\psi^a}^1(\omega_a)$ by (2.12). Indeed, this follows since $\tau_{hp} = v$ is continuous, with no face jumps, and since (crucial for variable $p$) the canonical elementwise Lagrange projection (2.12) employs the space $\mathcal{P}_{p_K}(K)$ from Section 2.7 with the polynomial degrees lowered according to the neighbors. Consequently, by (3.3a), $s_p^a = I_{hp}^L(\psi^a \tau_{hp})$. Then, (3.3b), the partition of unity (2.2), the linearity as well as the projection property of $I_{hp}^L$, and (2.10) give

$$s_{hp} = \sum_{a \in \mathcal{V}_h} s_p^a = \sum_{a \in \mathcal{V}_h} I_{hp}^L(\psi^a \tau_{hp}) = I_{hp}^L \left( \sum_{a \in \mathcal{V}_h} \psi^a \tau_{hp} \right) = I_{hp}^L(\tau_{hp}) = \tau_{hp}. \quad (5.1)$$

Thus, also $s_{hp} = v$. Next, step III. of Definition 3.3 builds on $\tau_{hp} - s_{hp}$, so that the bubble correction $\delta_{hp}$ is always zero (for any polynomial degree). Finally, step IV. builds on $\tau_{hp} - s_{hp} - \delta_{hp}$, so that all $\zeta_p^a$, $\zeta_p^{a,a}$, and $\zeta_{hp}$ are likewise zero. Then, from (3.8), $P_{hp}(v) = s_{hp} + \delta_{hp} + \zeta_{hp} = s_{hp} = v$. $\square$

## 5.2 Stability

Stability is a simple consequence of approximation (3.11):

**Lemma 5.2** (Stability property (3.10)). The stability property (3.10) holds true.

**Proof.** This follows by the triangle inequality from the approximation (3.11). Indeed, let $K \in \tilde{T}_h$ be fixed. Then

$$\|\nabla P_{hp}(v)\|_K^2 \le (\|\nabla v\|_K + \|\nabla(v - P_{hp}(v))\|_K)^2 \lesssim \sum_{L \in \tilde{T}_K} \|\nabla v\|_L^2,$$

where we have also used the trivial stability $\min_{v_p \in \mathcal{P}_{p_K}(L)} \|\nabla(v - v_p)\|_L \le \|\nabla v\|_L$. $\square$

<figure><img src="image_5.png" /></figure>
## 5.3 Approximation

We are left to prove the approximation property (3.11). We establish it first in the case of a uniform polynomial degree $p_K = p$ for all $K \in \mathcal{T}_h$ when $p \ge d+1$. Then we generalize it to the variable polynomial degree $p$.

**Lemma 5.3** (Approximation property (3.11), uniform polynomial degree $p \ge d+1$). *Consider a uniform polynomial degree $p_K = p \ge d+1$ for all $K \in \mathcal{T}_h$. Then the approximation property (3.11) holds true.*

*Proof.* Let $K \in \mathcal{T}_h$ be fixed. As $p \ge d+1$, formulas (3.4b), (3.5b), (3.6b) apply.

(i) Like in problem (3.5b), but on the extended element patch $\tilde{\mathcal{T}}_K$ in place of the extended vertex patch $\tilde{\mathcal{T}}_a$, see Section 2.2 and Figure 2 (right), define

$$ \zeta_p^K := \arg \min_{\substack{v_p \in \mathcal{P}_p(\tilde{\mathcal{T}}_K) \cap H_{0,D}^1(\tilde{\omega}_K) \\ (v_p, 1)_{\tilde{\omega}_K} = (\tau_{hp} - s_{hp} - \delta_{hp}, 1)_{\tilde{\omega}_K} = 0 \text{ if } (\partial \tilde{\omega}_K \cap \Gamma_D)^\circ = \emptyset}} \|\nabla(\tau_{hp} - s_{hp} - \delta_{hp} - v_p)\|_{\tilde{\omega}_K}. \quad (5.2) $$

To $\zeta_p^K$, we furthermore apply a mean value adjustment by elementwise bubble functions as in (3.4b), i.e., we define $\tilde{\delta}_{hp}^K \in \mathcal{P}_{d+1}(\tilde{\mathcal{T}}_K)$ by

$$ \tilde{\delta}_{hp}^K|_L := \frac{(\tau_{hp} - s_{hp} - \delta_{hp} - \zeta_p^K, 1)_L}{(b_L, 1)_L} b_L \quad \forall L \in \tilde{\mathcal{T}}_K, \quad (5.3) $$

and

$$ \begin{aligned} \tilde{\zeta}_p^K := \zeta_p^K + \tilde{\delta}_{hp}^K, \quad & (\tilde{\zeta}_p^K, 1)_L = (\zeta_p^K, 1)_L + \frac{(\tau_{hp} - s_{hp} - \delta_{hp} - \zeta_p^K, 1)_L}{(b_L, 1)_L} (b_L, 1)_L \\ & = (\tau_{hp} - s_{hp} - \delta_{hp}, 1)_L \stackrel{(4.1)}{=} 0 \quad \forall L \in \tilde{\mathcal{T}}_K, \end{aligned} \quad (5.4) $$

where we have crucially used that $\tau_{hp} - s_{hp} - \delta_{hp}$ is of elementwise mean value zero.

(ii) Importantly, the above bubble correction is $p$-stable in that

$$ \|\nabla(\tau_{hp} - s_{hp} - \delta_{hp} - \tilde{\zeta}_p^K)\|_{\tilde{\omega}_K} \lesssim \|\nabla(\tau_{hp} - s_{hp} - \delta_{hp} - \zeta_p^K)\|_{\tilde{\omega}_K} + \|\nabla(v - \tau_{hp})\|_{\tilde{\omega}_K}, \quad (5.5) $$

where the constant hidden in $\lesssim$ only depends on the shape-regularity of the mesh $\mathcal{T}_h$ and the space dimension $d$. Indeed, on the one hand,

$$ \begin{aligned} \|\nabla(\tau_{hp} - s_{hp} - \delta_{hp} - \tilde{\zeta}_p^K)\|_{\tilde{\omega}_K} &= \|\nabla(\tau_{hp} - s_{hp} - \delta_{hp} - \zeta_p^K - \tilde{\delta}_{hp}^K)\|_{\tilde{\omega}_K} \\ &\le \|\nabla(\tau_{hp} - s_{hp} - \delta_{hp} - \zeta_p^K)\|_{\tilde{\omega}_K} + \|\nabla \tilde{\delta}_{hp}^K\|_{\tilde{\omega}_K}. \end{aligned} $$

On the other hand, for any fixed $L \in \tilde{\mathcal{T}}_K$,

$$ \|\nabla b_L\|_L \lesssim h_L^{-1} \|b_L\|_L, \quad \|b_L\|_L |L|^{1/2} \approx (b_L, 1)_L \quad (5.6) $$

by the inverse inequality, equivalence of norms on finite-dimensional spaces, and mesh shape regularity; recall that $b_L$ is a low-order ($d+1$)-degree nonnegative polynomial, so that the constants hidden in $\lesssim$ and $\approx$ above are indeed independent of $p$. Thus, also using the fact that $(\tau_{hp}, 1)_L = (v, 1)_L$ from (3.2) and the Cauchy-Schwarz inequality,

$$ \begin{aligned} \|\nabla \tilde{\delta}_{hp}^K\|_L &= \frac{|(v - s_{hp} - \delta_{hp} - \zeta_p^K, 1)_L|}{(b_L, 1)_L} \|\nabla b_L\|_L \le \frac{\|v - s_{hp} - \delta_{hp} - \zeta_p^K\|_L |L|^{1/2}}{(b_L, 1)_L} \|\nabla b_L\|_L \\ &\lesssim h_{\tilde{\omega}_K}^{-1} \|v - s_{hp} - \delta_{hp} - \zeta_p^K\|_{\tilde{\omega}_K} \lesssim \|\nabla(v - s_{hp} - \delta_{hp} - \zeta_p^K)\|_{\tilde{\omega}_K} \\ &\le \|\nabla(\tau_{hp} - s_{hp} - \delta_{hp} - \zeta_p^K)\|_{\tilde{\omega}_K} + \|\nabla(v - \tau_{hp})\|_{\tilde{\omega}_K}; \end{aligned} $$

in the last but one step, we have applied the Poincaré inequality, since, crucially from (5.2) (recall $(\tau_{hp}, 1)_L = (v, 1)_L$ on $L$), $(v - s_{hp} - \delta_{hp} - \zeta_p^K) \in H^1(\tilde{\omega}_K)$ is zero on $\partial \tilde{\omega}_K \cap \Gamma_D$ or of mean value zero on $\tilde{\omega}_K$.

(iii) Now, as in (3.6b), we decompose the mean-value adjusted $\tilde{\zeta}_p^K$ from (5.4) using Corollary A.1 (with $\mathcal{T}_\omega = \tilde{\mathcal{T}}_K$ and $\mathcal{V}_\omega = \tilde{\mathcal{V}}_K$)

$$ \tilde{\zeta}_p^K = \sum_{b \in \tilde{\mathcal{V}}_K} \tilde{\zeta}_p^{K,b} \quad \text{with} \quad \tilde{\zeta}_p^{K,b} \in \mathcal{P}_p(\tilde{\mathcal{T}}_b \cap \tilde{\mathcal{T}}_K) \cap H_{0,D,\psi^b}^1(\omega_b, \tilde{\omega}_K). \quad (5.7) $$
Assumptions (A.1) are indeed satisfied by $\tilde{\zeta}_p^K$; in particular, the boundary condition comes from the constraint in (5.2) (the shift from $\zeta_p^K$ to $\tilde{\zeta}_p^K$ is by bubbles that are zero on the mesh faces) and the mean value condition comes from (5.4) (mean value zero on all elements $L \in \tilde{T}_K$ implies mean value zero on $\tilde{\omega}_K$). Now, crucially, as in (3.6b), the contributions for the vertices $a$ of the element $K$, $a \in V_K$, lie in $\mathcal{P}_p(\mathcal{T}_a) \cap H_{0,D,\psi^a}^1(\omega_a)$ (as $\mathcal{T}_a$ are included in $\tilde{T}_K$, $\mathcal{T}_a \cap \tilde{T}_K = \mathcal{T}_a$ and no patch truncation happens).

(iv) For each vertex $a \in V_K$, let us also consider $\tilde{\zeta}_p^K$ from (5.4) restricted to the extended vertex patch $\tilde{\omega}_a$ (recall that $\tilde{\omega}_a \subset \tilde{\omega}_K$ by definition, see Figure 2, right). We again decompose $\tilde{\zeta}_p^K|_{\tilde{\omega}_a}$ using Corollary A.1 (this time with $\mathcal{T}_\omega = \tilde{T}_a$ and $V_\omega = \tilde{V}_a$)

$$ \tilde{\zeta}_p^K|_{\tilde{\omega}_a} = \sum_{b \in \tilde{V}_a} \tilde{\zeta}_p^{K,a,b} \quad \text{with} \quad \tilde{\zeta}_p^{K,a,b} \in \mathcal{P}_p(\tilde{T}_b \cap \tilde{T}_a) \cap H_{0,D,\psi^b}^1(\omega_b, \tilde{\omega}_a). \qquad (5.8) $$

Again, $\tilde{\zeta}_p^K|_{\tilde{\omega}_a}$ satisfy assumptions (A.1). This would not be the case of $\zeta_p^K$ from (5.2), which only satisfies the mean value condition (A.1b) on the extended element patch $\tilde{\omega}_K$ but not on its subpatches $\tilde{\omega}_a$; for this reason, we pass from $\zeta_p^K$ to $\tilde{\zeta}_p^K$.

(v) Now a central observation comes from (A.3): since $\tilde{\zeta}_p^K$ and $\tilde{\zeta}_p^K|_{\tilde{\omega}_a}$ are identical on the extended patches $\tilde{\omega}_a$, the $d+1$ contributions $\tilde{\zeta}_p^{K,a}$ from (5.7) for the vertices $a$ of the element $K$ respectively coincide with the $d+1$ contributions $\tilde{\zeta}_p^{K,a,a}$ from (5.8),

$$ \tilde{\zeta}_p^{K,a} = \tilde{\zeta}_p^{K,a,a} \quad \forall a \in V_K. \qquad (5.9) $$

Indeed, by (A.3), these contributions have the vertex patches $\mathcal{T}_a$ as support and the extended vertex patches $\tilde{\mathcal{T}}_a$ as dependency regions. This is actually the reason for the remainder potential reconstruction (3.5b) and the decomposition (3.6b) to be performed on the extended vertex patches $\tilde{\mathcal{T}}_a$; merely the vertex patches $\mathcal{T}_a$ would not be sufficient, as the dependency regions are not given by the vertex patches $\mathcal{T}_a$ but by the extended vertex patches $\tilde{\mathcal{T}}_a$. Overall, from (5.7)–(5.9), we conclude that

$$ \tilde{\zeta}_p^K|_{K} \stackrel{(5.7)}{=} \sum_{a \in V_K} \tilde{\zeta}_p^{K,a}|_K \stackrel{(5.9)}{=} \sum_{a \in V_K} \tilde{\zeta}_p^{K,a,a}|_K. \qquad (5.10) $$

(vi) Recall now the definition of $\tau_{hp}$ from (3.2) and that $K \in \mathcal{T}_h$ is fixed. We estimate by the triangle inequality and employing the definitions (3.8) and (3.7) together with the equality (5.10),

$$ \begin{align} & \|\nabla(v - P_{hp}(v))\|_K \tag{5.11} \\ & \le \|\nabla(v - \tau_{hp})\|_K + \|\nabla(\tau_{hp} - s_{hp} - \delta_{hp} - \zeta_{hp})\|_K \tag{3.8} \\ & \stackrel{(3.7)}{=} \|\nabla(v - \tau_{hp})\|_K + \left\|\nabla\left(\tau_{hp} - s_{hp} - \delta_{hp} - \zeta_p^K + \sum_{a \in V_K} (\zeta_p^{K,a,a} - \zeta_p^{a,a})\right)\right\|_K \tag{5.10} \\ & \lesssim \|\nabla(v - \tau_{hp})\|_{\tilde{\omega}_K} + \|\nabla(\tau_{hp} - s_{hp} - \delta_{hp} - \zeta_p^K)\|_{\tilde{\omega}_K} + \sum_{a \in V_K} \|\nabla(\zeta_p^{K,a,a} - \zeta_p^{a,a})\|_{\omega_a}; \tag{5.5} \end{align} $$

in the last estimate, we have crucially used (5.5) to go back from the bubble-shifted $\tilde{\zeta}_p^K$ to $\zeta_p^K$. From (3.2), the first term above already has the form requested in (3.11). For the last term, we crucially use the linearity of the decomposition (A.3) and its p-robust stability expressed by (A.5) from Corollary A.1 (note that $h_{\tilde{\omega}_K}/\min_{L \in \tilde{T}_K} h_L$ only depends on $\max_{L \in \tilde{T}_K} \kappa_L$). Thus, for a vertex $a \in V_K$, recalling (5.8) and (3.6b),

$$ \begin{align} & \|\nabla(\tilde{\zeta}_p^{K,a,a} - \zeta_p^{a,a})\|_{\omega_a} \tag{A.5} \\ & \lesssim \|\nabla(\tilde{\zeta}_p^K - \zeta_p^a)\|_{\tilde{\omega}_a} \nonumber \\ & \le \|\nabla(\tau_{hp} - s_{hp} - \delta_{hp} - \tilde{\zeta}_p^K)\|_{\tilde{\omega}_K} + \|\nabla(\tau_{hp} - s_{hp} - \delta_{hp} - \zeta_p^a)\|_{\tilde{\omega}_a} \tag{5.5} \\ & \lesssim \|\nabla(\tau_{hp} - s_{hp} - \delta_{hp} - \zeta_p^K)\|_{\tilde{\omega}_K} + \|\nabla(\tau_{hp} - s_{hp} - \delta_{hp} - \zeta_p^a)\|_{\tilde{\omega}_a} + \|\nabla(v - \tau_{hp})\|_{\tilde{\omega}_K}, \nonumber \end{align} $$

where we have added and subtracted $\tau_{hp} - s_{hp} - \delta_{hp}$, used the triangle inequality, extended the integration region, and again used (5.5). We are thus left estimating $\|\nabla(\tau_{hp} - s_{hp} - \delta_{hp} - \zeta_p^a)\|_{\tilde{\omega}_a}$ for $\zeta_p^a$ given by (3.5b)
and $\|\nabla(\tau_{hp} - s_{hp} - \delta_{hp} - \zeta_p^K)\|_{\tilde{\omega}_K}$ for $\zeta_p^K$ given by (5.2). As these take the same form, we only show details for the latter.

(vii) Let us finally consider $\|\nabla(\tau_{hp} - s_{hp} - \delta_{hp} - \zeta_p^K)\|_{\tilde{\omega}_K}$ with $\zeta_p^K$ given by (5.2). Such problems (recall that $\tau_{hp}$ from (3.2) is merely a discontinuous piecewise polynomial from $\mathcal{P}_p(\tilde{T}_h)$ but not from $H_{0,D}^1(\Omega)$) have recently been analyzed and p-robust stability has been shown in [23, Corollaries 3.1 and 3.7], see also [13, Corollaries 3.4 and 4.4] on: 1) vertex patch subdomains $\omega_a$; and 2) with Dirichlet boundary conditions on $\partial\omega_a$. We now rely on the extension of this result to larger (extended) patch subdomains where Dirichlet boundary conditions are not prescribed from Appendix C. As a preliminary step, though, first note that

$$ \begin{aligned} \|\nabla(\tau_{hp} - s_{hp} - \delta_{hp} - \zeta_p^K)\|_{\tilde{\omega}_K} &= \min_{v_p \in \mathcal{P}_p(\tilde{T}_K) \cap H_{0,D}^1(\tilde{\omega}_K)} \|\nabla(\tau_{hp} - s_{hp} - \delta_{hp} - v_p)\|_{\tilde{\omega}_K} \\ &= \min_{v_p \in \mathcal{P}_p(\tilde{T}_K) \cap H_{0,D}^1(\tilde{\omega}_K)} \|\nabla(\tau_{hp} - v_p)\|_{\tilde{\omega}_K}. \end{aligned} \quad (5.12) $$

Indeed, this follows by the shift by $(s_{hp} + \delta_{hp})|_{\tilde{\omega}_K}$ since, by (3.3b), it lies in $\mathcal{P}_p(\tilde{T}_K) \cap H_{0,D}^1(\tilde{\omega}_K)$; the (trace) continuity of $s_{hp} + \delta_{hp}$ together with $s_{hp} + \delta_{hp} = 0$ on $\Gamma_D$ are crucial here. In this important conceptual step, the non p-robust usual potential reconstruction $s_{hp}$, together with its mean value correction $\delta_{hp}$, are played out. Please note that the minimum in (5.2) and that in the above first line coincide: the mean value constraint in (5.2) only fixes $\zeta_p^K$ uniquely but has no influence on its gradient.

We now finally apply Corollary C.2 on the extended vertex patch $\tilde{T}_K$ (recall that we suppose a uniform polynomial degree $p$ here) to deduce that

$$ \min_{v_p \in \mathcal{P}_p(\tilde{T}_K) \cap H_{0,D}^1(\tilde{\omega}_K)} \|\nabla(\tau_{hp} - v_p)\|_{\tilde{\omega}_K} \lesssim \min_{w \in H_{0,D}^1(\tilde{\omega}_K)} \|\nabla(\tau_{hp} - w)\|_{\tilde{\omega}_K}. \quad (5.13) $$

Finally, we can play in the target function $v \in H_{0,D}^1(\Omega)$ from the announcement of Theorem 3.4, which satisfies $v|_{\tilde{\omega}_K} \in H_{0,D}^1(\tilde{\omega}_K)$, and obtain

$$ \min_{w \in H_{0,D}^1(\tilde{\omega}_K)} \|\nabla(\tau_{hp} - w)\|_{\tilde{\omega}_K} \lesssim \|\nabla(\tau_{hp} - v)\|_{\tilde{\omega}_K}. \quad (5.14) $$

Combining the above bounds (5.11)-(5.14) together with the definition (3.2) of $\tau_{hp}$ gives the assertion (3.11). □

**Lemma 5.4** (Approximation property (3.11), variable polynomial degree). Consider the general case of a variable polynomial degree $p$. Then the approximation property (3.11) holds true.

*Proof.* Let $K \in \tilde{T}_h$ be fixed. We consider several cases.

(i) Case $p_K < d+1$. In this low-polynomial-degree case, following (3.4a), there is no bubble correction, $\delta_{hp}|_K = 0$. Moreover, on each extended vertex patch $\tilde{T}_a$ for $a \in \mathcal{V}_h$ such that $K \in \tilde{T}_a$, by (3.5a) and (3.6a), there holds $\zeta_p^a = \zeta_p^{a,a} = 0$, so that, in view of (3.7), there is also no p-robust correction, $\zeta_{hp}|_K = 0$. Thus, by (3.8), $(P_{hp}(v))|_K = s_{hp}|_K$, where $s_{hp}$ is given by (3.3). This is the usual construction of $s_{hp}$ from [22, Construction 3.8 and Remark 3.10] and [23, Corollaries 3.1 and 3.7 together with equation (4.3)]. As discussed in point 2 in Section 3.1.2, it is not p-robust, but here the polynomial degree is low, $p_K < d+1$, so there will be no problem. To see (3.11), we start by the triangle inequality,

$$ \|\nabla(v - P_{hp}(v))\|_K = \|\nabla(v - s_{hp})\|_K \le \|\nabla(v - \tau_{hp})\|_K + \|\nabla(\tau_{hp} - s_{hp})\|_K. $$

Then, similarly to the above references and using the three last equalities from (5.1) together with (3.3b), we see

$$ \|\nabla(\tau_{hp} - s_{hp})\|_K = \left\| \nabla \left( \sum_{a \in \mathcal{V}_K} (I_{hp}^L(\psi^a \tau_{hp}) - s_p^a) \right) \right\|_K \le \sum_{a \in \mathcal{V}_K} \|\nabla(I_{hp}^L(\psi^a \tau_{hp}) - s_p^a)\|_{\omega_a}. $$

If the polynomial degree in the neighbors of $K$ is different from $p_K$, then the use of the polynomial space $\mathcal{P}_{p_K}(K)$ with degree lowered according to neighbors from Section 2.7 in (3.2) is crucial, since only then, cf. (3.2), $\tau_{hp} \in \mathcal{P}_p(\tilde{T}_h)$, and only then $I_{hp}^L(\tau_{hp}) = \tau_{hp}$, which is crucially used in (5.1).

For each vertex $a \in \mathcal{V}_K$, let us next introduce the following auxiliary problem:

$$ \hat{s}_p^a := \arg \min_{v_p \in \mathcal{P}_{p+1}(\tilde{T}_a) \cap H_{0,D,\psi a}^1(\omega_a)} \|\nabla(\psi^a \tau_{hp} - v_p)\|_{\omega_a}. $$
It is similar to (3.3a), but increases the polynomial degree by one and does not employ the elementwise Lagrange projector $I_{hp}^L$. We observe that $I_{hp}^L(\hat{s}_p^a) \in \mathcal{P}_p(T_a) \cap H_{0,D,\psi^a}(\omega_a)$. Thus, since $s_p^a$ is the minimizer from (3.3a) and since $I_{hp}^L$ is linear,

$$ \|\nabla(I_{hp}^L(\psi^a \tau_{hp}) - s_p^a)\|_{\omega_a} \le \|\nabla(I_{hp}^L(\psi^a \tau_{hp} - \hat{s}_p^a))\|_{\omega_a}. \quad (5.15) $$

Finally, we employ the stability of $I_{hp}^L$ (not p-robust but fine here since $p_K < d+1$ and since the variation of the polynomial degree in the patch $T_a$ is controlled by $\max_{L \in T_a} \kappa_{pL}$ of (2.8)). Employing also [23, Corollaries 3.1 and 3.7 together with equation (4.3)] (for the given function $v \in H_{0,D}^1(\Omega)$, $\psi^a v \in H_{0,D,\psi^a}^1(\omega_a)$), we have

$$ \|\nabla(I_{hp}^L(\psi^a \tau_{hp} - \hat{s}_p^a))\|_{\omega_a} \lesssim \|\nabla(\psi^a \tau_{hp} - \hat{s}_p^a)\|_{\omega_a} \lesssim \|\nabla(\psi^a(\tau_{hp} - v))\|_{\omega_a}. $$

Finally,

$$ \begin{aligned} \|\nabla(\psi^a(\tau_{hp} - v))\|_{\omega_a}^2 &= \sum_{L \in T_a} \|\nabla\psi^a(\tau_{hp} - v) + \psi^a\nabla(\tau_{hp} - v)\|_L^2 \\ &\le \sum_{L \in T_a} \{\|\nabla\psi^a\|_{\infty,L}\|\tau_{hp} - v\|_L + \|\psi^a\|_{\infty,L}\|\nabla(\tau_{hp} - v)\|_L\}^2 \\ &\lesssim \|\nabla(\tau_{hp} - v)\|_{\omega_a}^2, \end{aligned} $$

applying $\|\nabla\psi^a\|_{\infty,L} \lesssim h_L^{-1}$, $\|\psi^a\|_{\infty,L} = 1$, and the Poincaré inequality $\|\tau_{hp} - v\|_L \le h_L/\pi \|\nabla(\tau_{hp} - v)\|_L$, since $(\tau_{hp}, 1)_L = (v, 1)_L$ for all $L \in T_a$ from the constraint in (3.2), similarly to, e.g., [22, inequality (3.29)]. Thus, by combining the above inequalities,

$$ \|\nabla(v - P_{hp}(v))\|_K \lesssim \|\nabla(v - \tau_{hp})\|_{\omega_K}, $$

which implies (3.11).

(ii) Case $p_K \ge d+1$ and $\zeta_p^a = \zeta_p^{a,a} = 0$ for all $a \in \mathcal{V}_h$ such that $K \in \tilde{T}_a$, so that $\zeta_{hp}|_K = 0$, resulting from $p_L < d+1$ for some $L \in \tilde{T}_a$ and the prescriptions (3.5a), (3.6a). Consequently, in view of (2.8), $p_K$ is still small in that it can be controlled by the space dimension $d$ and the polynomial-variation parameter $\max_{L \in \tilde{T}_K} \kappa_{pL}$ from (2.8). Here, by (3.8), $(P_{hp}(v))|_K = s_{hp}|_K + \delta_{hp}|_K$, where $s_{hp}$ is given by (3.3) and $\delta_{hp}$ by (3.4b). As in the proof of Lemma 5.3, step (i),

$$ \begin{aligned} \|\nabla\delta_{hp}\|_K &\stackrel{(3.4b)}{\le} \frac{\|\tau_{hp} - s_{hp}\|_K |K|^{1/2}}{(b_K, 1)_K} \|\nabla b_K\|_K \stackrel{(5.6)}{\lesssim} h_K^{-1} \|\tau_{hp} - s_{hp}\|_K \\ &\stackrel{(5.1)}{=} h_K^{-1} \left\| \sum_{a \in \mathcal{V}_K} (I_{hp}^L(\psi^a \tau_{hp}) - s_p^a) \right\|_K \le \sum_{a \in \mathcal{V}_K} \|\nabla(I_{hp}^L(\psi^a \tau_{hp}) - s_p^a)\|_K; \end{aligned} $$

note that $I_{hp}^L(\psi^a \tau_{hp}) - s_p^a$ is zero on at least one face of $K$, which allows us to use the Poincaré inequality in the last step. Thus, the triangle inequality

$$ \|\nabla(v - P_{hp}(v))\|_K = \|\nabla(v - s_{hp} - \delta_{hp})\|_K \le \|\nabla(v - \tau_{hp})\|_K + \|\nabla(\tau_{hp} - s_{hp})\|_K + \|\nabla\delta_{hp}\|_K $$

in combination with step (i) above, branching at (5.15), implies

$$ \|\nabla(v - P_{hp}(v))\|_K \lesssim \|\nabla(v - \tau_{hp})\|_{\omega_K} $$

and consequently (3.11) also in this case.

(iii) Case $p_K \ge d+1$ and $\zeta_p^a, \zeta_p^{a,a}$ non (necessarily) zero for all $a \in \mathcal{V}_h$ such that $K \in \tilde{T}_a$, resulting from the prescriptions (3.5b), (3.6b). Note that consequently $p_L \ge d+1$ for all $L \in \tilde{T}_K$, and we are in a high-polynomial-degree case. The proof of Lemma 5.3 applies verbatim until (5.12); in particular, the mean value fixes (5.4) are well defined for all $L \in \tilde{T}_K$, since $p_L \ge d+1$, and variable polynomial degree is treated in Corollary A.1. We cannot, unfortunately, proceed as in the last paragraph, namely for (5.13), since Corollary C.2 only addresses uniform polynomial degree. We thus adjust the arguments as follows. In addition to $\tau_{hp}$ from (3.2), let, for all $L \in \tilde{T}_K$,

$$ \tau_{hp}^K|_L := \arg \min_{\substack{v_p \in \mathcal{P}_{\tilde{T}_K}(L) \\ (v_p, 1)_L = (v, 1)_L}} \|\nabla(v - v_p)\|_L, \quad (5.16) $$
where we recall that $\underline{p}_K$ is the smallest polynomial degree over the extended element patch $\tilde{T}_K$, see (2.7). Here, $\tau_{hp}^K$ is a lowered uniform polynomial degree local-best approximate of $v$, $\tau_{hp}^K \in \mathcal{P}_{\underline{p}_K}(\tilde{T}_K)$. Let us also denote the argument of the minimization on the left-hand side of (5.13) by $\iota_p^K$,

$$ \begin{array}{ll} \iota_p^K := \arg \min_{\substack{v_p \in \mathcal{P}_p(\tilde{T}_K) \cap H_{0,D}^1(\tilde{\omega}_K) \\ (v_p, 1)_{\tilde{\omega}_K} = (s_{hp} + \delta_{hp}, 1)_{\tilde{\omega}_K} \text{ if } (\partial \tilde{\omega}_K \cap \Gamma_D)^\circ = \emptyset}} & \|\nabla(\tau_{hp} - v_p)\|_{\tilde{\omega}_K}; \tag{5.17} \\ & \|\nabla(\tau_{hp} - v_p)\|_{\tilde{\omega}_K}; \end{array} $$

note that from (5.2) and (5.12), $\iota_p^K = \zeta_p^K + (s_{hp} + \delta_{hp})|_{\tilde{\omega}_K}$. Then, also consider

$$ \begin{array}{ll} \iota_p^K := \arg \min_{\substack{v_p \in \mathcal{P}_{\underline{p}_K}(\tilde{T}_K) \cap H_{0,D}^1(\tilde{\omega}_K) \\ (v_p, 1)_{\tilde{\omega}_K} = (s_{hp} + \delta_{hp}, 1)_{\tilde{\omega}_K} \text{ if } (\partial \tilde{\omega}_K \cap \Gamma_D)^\circ = \emptyset}} & \|\nabla(\tau_{hp}^K - v_p)\|_{\tilde{\omega}_K}; \tag{5.18} \\ & \|\nabla(\tau_{hp}^K - v_p)\|_{\tilde{\omega}_K}; \end{array} $$

this is a lowered uniform polynomial degree version of (5.17) which employs $\tau_{hp}^K$ from (5.16) in place of $\tau_{hp}$ from (3.2).

Since $\iota_p^K$ belongs to the minimization set in (5.17), the triangle inequality, Corollary C.2 (applied with the uniform lowered polynomial degree $\underline{p}_K$), and the trivial estimate $\|\nabla(\tau_{hp} - v)\|_L \le \|\nabla(\tau_{hp}^K - v)\|_L$ for all $L \in \tilde{T}_K$ give, in replacement of (5.13),

$$ \begin{align*} \min_{\substack{v_p \in \mathcal{P}_p(\tilde{T}_K) \cap H_{0,D}^1(\tilde{\omega}_K) \\ (v_p, 1)_{\tilde{\omega}_K} = (s_{hp} + \delta_{hp}, 1)_{\tilde{\omega}_K} \text{ if } (\partial \tilde{\omega}_K \cap \Gamma_D)^\circ = \emptyset}} & \|\nabla(\tau_{hp} - v_p)\|_{\tilde{\omega}_K} \\ & = \|\nabla(\tau_{hp} - \iota_p^K)\|_{\tilde{\omega}_K} \le \|\nabla(\tau_{hp} - \iota_p^K)\|_{\tilde{\omega}_K} \\ & \le \|\nabla(\tau_{hp} - v)\|_{\tilde{\omega}_K} + \|\nabla(\tau_{hp}^K - v)\|_{\tilde{\omega}_K} + \|\nabla(\tau_{hp}^K - \iota_p^K)\|_{\tilde{\omega}_K} \\ & \lesssim \|\nabla(\tau_{hp} - v)\|_{\tilde{\omega}_K} + \|\nabla(\tau_{hp}^K - v)\|_{\tilde{\omega}_K} + \min_{w \in H_{0,D}^1(\tilde{\omega}_K)} \|\nabla(\tau_{hp}^K - w)\|_{\tilde{\omega}_K} \\ & \le \|\nabla(\tau_{hp} - v)\|_{\tilde{\omega}_K} + \|\nabla(\tau_{hp}^K - v)\|_{\tilde{\omega}_K} + \|\nabla(\tau_{hp}^K - v)\|_{\tilde{\omega}_K} \le 3 \|\nabla(\tau_{hp}^K - v)\|_{\tilde{\omega}_K}. \end{align*} $$

We have also played in the target function $v \in H_{0,D}^1(\Omega)$ from the announcement of Theorem 3.4. Thus, (3.11) follows from the definition (5.16).

(iv) Case $p_K \ge d+1$ and $\zeta_p^b = \zeta_p^{b,b} = 0$ for some $b \in \mathcal{V}_h$ such that $K \in \tilde{T}_b$, resulting from the prescriptions (3.5a), (3.6a) since $p_L < d+1$ for some $L \in \tilde{T}_b$, whereas $\zeta_p^a, \zeta_p^{a,a}$ non (necessarily) zero for some other $a \in \mathcal{V}_h$ such that $K \in \tilde{T}_a$, resulting from the prescriptions (3.5b), (3.6b). Here, by (3.8), $(P_{hp}(v))|_K = s_{hp}|_K + \delta_{hp}|_K + \sum_{a \in \hat{\mathcal{V}}_K} \zeta_p^{a,a}$, where $s_{hp}$ is given by (3.3), $\delta_{hp}$ is given by (3.4b), $\zeta_p^{a,a}$ is given by (3.5b), (3.6b), and $\hat{\mathcal{V}}_K$ collects the vertices of the element $K$ with $\zeta_p^a, \zeta_p^{a,a}$ prescribed by (3.5b), (3.6b). We cannot proceed exactly as in the above step (iii): the problem (5.2) is well-defined, but the elementwise mean values (5.4) cannot be fixed for those $L \in \tilde{T}_b \cap \tilde{T}_K$ where the polynomial degree is too small, $p_L < d+1$.

The fix is, fortunately, easy. Define $\hat{T}_K := \cup_{a \in \hat{\mathcal{V}}_K} \{\tilde{T}_a\}$, which is a subset of the extended element patch $\tilde{T}_K$; the corresponding subdomain $\hat{\omega}_K$ is then a subset of $\tilde{\omega}_K$. We then introduce $\zeta_p^K$ as in (5.2) but restricting the minimization to $\hat{T}_K$ and $\hat{\omega}_K$. Then the mean value fixes (5.4) are well defined for all $L \in \hat{T}_K$ and one can verify that the proof proceeds employing the reduced set of vertices $\hat{\mathcal{V}}_K$ in place of $\mathcal{V}_K$. $\square$

## A A $p$-stable $\mathcal{P}_p \cap H^1$ decomposition on patches

We now state a $p$-stable decomposition result which follows from Schöberl et al. [32] for a uniform polynomial degree and $d=2,3$ and, for $d=2$, Karkulik et al. [28] for a variable polynomial degree. Recall the notation from Sections 2.6 and 2.7.

**Corollary A.1 (A $p$-stable $\mathcal{P}_p \cap H^1$ decomposition on patches).** Let a simplicial mesh $\mathcal{T}_h$ of $\Omega$, a polynomial degree $p$, and a simplicial submesh $\mathcal{T}_\omega$ of $\mathcal{T}_h$ with the corresponding open, bounded, and Lipschitz polygon or polyhedron $\omega \subset \mathbb{R}^d$, $d=2,3$, with $\bar{\omega}$ contractible. Let $\mathcal{V}_\omega$ be a vertex set of $\mathcal{T}_\omega$. For $d=3$ and variable polynomial degree, let Assumption 3.2 hold. Let

$$ s_p \in \mathcal{P}_p(\mathcal{T}_\omega) \cap H_{0,D}^1(\omega), \tag{A.1a} $$

$$ (s_p, 1)_\omega = 0 \text{ if } (\partial\omega \cap \Gamma_D)^\circ = \emptyset \tag{A.1b} $$
be a **p**-degree continuous piecewise polynomial on $\omega$, respecting the zero trace condition on $\Gamma_D$, or mean-value free. Then there exists a decomposition of $s_p$ as

$$s_p = \sum_{b \in \mathcal{V}_\omega} s_p^b \qquad (A.2)$$

where the contributions

$s_p^b$ are supported on the vertex patch subdomains $\omega_b \cap \omega$, linearly depend on $s_p$ on the extended vertex patch subdomains $\tilde{\omega}_b \cap \omega$, \qquad (A.3)

and satisfy

$$s_p^b \in \mathcal{P}_p(\mathcal{T}_b \cap \mathcal{T}_\omega) \cap H_{0,D,\psi^b}^1(\omega_b, \omega), \qquad (A.4)$$

i.e., recalling (2.9) and (2.5), $s_p^b$ are **p**-degree continuous piecewise polynomial on $\omega_b \cap \omega$ and $s_p^b = 0$ on those faces in $\partial(\omega_b \cap \omega)$ where the hat function $\psi^b$ vanishes or which lie on the Dirichlet boundary $\bar{\Gamma}_D$. Moreover, the decomposition is **p**-stable as

$$\|\nabla s_p^b\|_{\omega_b \cap \omega} \lesssim \|\nabla s_p\|_{\tilde{\omega}_b \cap \omega} \quad \forall b \in \mathcal{V}_\omega, \qquad (A.5)$$

where the constant hidden in $\lesssim$ only depends on the local mesh shape-regularity parameter $\kappa_{\mathcal{T}_\omega}$ given by $\kappa_{\mathcal{T}_\omega} := \max_{K \in \mathcal{T}_\omega} \kappa_K$ and the ratio $h_\omega / \min_{K \in \mathcal{T}_\omega} h_K$.

**Remark A.2** (No global low order component and no local orthogonality constraints). In stable compositions, there usually appears a lowest-order component supported over the whole domain $\omega$ (as in Schöberl et al. [32, Section 3.1]), or some local orthogonality constraints are imposed (as in [12, Appendix B]). This is avoided here under the zero trace or mean value condition in (A.1) and for the price of the ratio $h_\omega / \min_{K \in \mathcal{T}_\omega} h_K$. Note that, for $\mathcal{T}_\omega$ the local patches $\tilde{\mathcal{T}}_a$ or $\tilde{\mathcal{T}}_K$, this ratio is bounded solely in function of the mesh shape regularity parameter $\kappa_h$ from (2.1).

*Proof*, uniform polynomial degree $p_K = p \ge 1$ for all $K \in \mathcal{T}_h$. Let $s_p$ satisfy (A.1) and consider the decomposition of Schöberl et al. [32] on the domain $\omega$ and mesh $\mathcal{T}_\omega$. Importantly, we choose the “coarse grid contribution” ($u_0$ in equation (2) or $\Pi_h u$ in Section 4.1 of [32]) as zero. This is eligible in terms of [32, Lemma 3.1], since

$$\|\nabla 0\|_{\omega} \le \|\nabla s_p\|_{\omega}, \qquad (A.6a)$$

$$\|\nabla s_p\|_{\omega} = \|\nabla s_p\|_{\omega}, \qquad (A.6b)$$

$$\|h^{-1}s_p\|_{\omega} \le \frac{1}{\min_{K \in \mathcal{T}_{\omega}} h_K} \|s_p\|_{\omega} \stackrel{\text{Poincaré}}{\lesssim} \frac{h_{\omega}}{\min_{K \in \mathcal{T}_{\omega}} h_K} \|\nabla s_p\|_{\omega}. \qquad (A.6c)$$

Consequently, there is no global low order component. The construction of [32] then gives the decomposition (A.2), see equations (11) and (24) (after associating the face, edge, and element contributions with the vertex contributions), **p**-stable as

$$\sum_{b \in \mathcal{V}_\omega} \|\nabla s_p^b\|_{\omega_b \cap \omega}^2 \lesssim \|\nabla s_p\|_{\omega}^2.$$

The inspection of the developments of [32] shows that $s_p^b$ are supported on $\omega_b \cap \omega$ and solely constructed from and linearly dependent on the values of $s_p$ on the extended patches $\tilde{\omega}_b \cap \omega$ as expressed in (A.3) and satisfy more precisely the local stability (A.5). Crucially, the constant hidden in $\lesssim$ above only depends on the shape-regularity parameter $\kappa_{\mathcal{T}_\omega}$ of the mesh $\mathcal{T}_\omega$ and, through (A.6c), on the ratio $h_\omega / \min_{K \in \mathcal{T}_\omega} h_K$. $\square$

*Proof*, variable polynomial degrees $p_K \ge 1$ for $K \in \mathcal{T}_h$. The proof is as above, relying, in two space dimensions, on Karkulik et al. [28, Theorem 2.5], and, in three space dimensions, on Assumption 3.2. $\square$

# B Shellability of patches of mesh elements

For the proofs of Appendix C below, we will need a suitable enumeration of patch simplices. For this purpose, we will use the concept of shellability of simplicial complexes from algebraic topology/discrete geometry. In our setting, (a pure) simplicial complex $\mathcal{T}_\omega$ is a simplicial mesh $\mathcal{T}_\omega$ where we also consider all the sub-simplices (vertices, edges, and, in three space dimensions, faces). Following Ziegler [37, Definition 8.1 and Remarks 8.3.(ii)], we define:
<figure><img src="image_6.png" /><figcaption>Figure 4: Examples of enumerations that are not shellings according to Definition B.1. The already enumerated simplices are dotted the simplex breaking the shellability is hatched.</figcaption></figure>

**Definition B.1** (Shelling of a simplicial complex). Let $\omega$ be an open and bounded polygon or polyhedron with $\bar{\omega}$ contractible. A shelling of a simplicial complex $\mathcal{T}_{\omega}$ with domain $\bar{\omega}$ is an enumeration $K_1 \dots K_{|\mathcal{T}_{\omega}|}$ of the $d$-dimensional simplices of $\mathcal{T}_{\omega}$ such that for all $1 < i \le |\mathcal{T}_{\omega}|$, the intersection of $K_i$ with the previously enumerated $d$-dimensional simplices is a nonempty collection of $(d-1)$-dimensional faces of $K_i$.

Definition B.1 means that the intersection of $K_i$ with the previously enumerated $d$-dimensional simplices cannot be and cannot include mere points (if $d=2$) and cannot be and cannot include mere points or edges (if $d=3$). Illustrations of enumerations that are not shellings are provided in Figure 4: the triangle $K_3$ only shares a point with the previously enumerated triangles $K_1$ and $K_2$ (left), the tetrahedron $K_2$ only shares an edge with the tetrahedron $K_1$ (middle), and the tetrahedron $K_5$ shares a face with the tetrahedron $K_4$ but a mere edge with the tetrahedron $K_1$ (right). Denote by $\mathcal{T}_{\omega,i}$ the simplicial complex formed by the $d$-dimensional simplices enumerated before $K_i$, $K_1, \dots, K_{i-1}$, $1 \le i \le |\mathcal{T}_{\omega}|$. Shellability gives that the closure of the underlying open domain of $\mathcal{T}_{\omega,i}$ is a triangulated manifold with boundary, more precisely a topological $d$-ball, for any $1 \le i \le |\mathcal{T}_{\omega}|$, see Ziegler [36, proof of Proposition 2.4.(iv)].

A simplicial complexes is shellable if it has at least one shelling. In two space dimensions, cf. Ziegler [37, Examples 8.4.(i)], the situation is simple:

**Theorem B.2** (Shellability of simplicial complexes for $d=2$). *All simplicial complexes over an open and bounded polygon $\omega$ in $\mathbb{R}^2$ with $\bar{\omega}$ contractible are shellable.*

In three space dimensions, a typical patch $\mathcal{T}_{\omega}$ from finite element mesh will also be shellable. However, rigorously, the situation is much more complex:

**Remark B.3** (Shellable simplicial complexes for $d=3$). *There holds:*

* Any simplicial complex with at most 8 vertices is shellable, see Lutz [29, Corollary 6]. (Conjecture: any complex with at most 17 tetrahedra is shellable).

* There exists a nonshellable simplicial complex with 9 vertices and 18 tetrahedra, see Lutz [29]. A nonshellable simplicial complex with 10 vertices and 21 tetrahedra with a graphical visualization is given in Ziegler [36].

* Any Delaunay simplicial mesh (and more generally any regular simplicial mesh) is shellable, see Ziegler [37, Definition 5.3 and Corollary 8.14].

* For every simplicial mesh, there exists a shellable subdivision, see Bruggesser and Mani [9, Proposition 1]. Actually, Adiprasito and Benedetti [1, Theorem A] show that barycentric refinement (inserting a barycenter to each edge, face, and tetrahedron) is sufficient.

## C $p$-stable broken $H^1$ polynomial extension on patch subdomains

We summarize here our results on local $p$-stable broken $H^1$ polynomial extensions. We only consider a uniform polynomial degree $p$.
## C.1 Available results on a single mesh element

p-stable $H^1$ polynomial extensions on a single triangle or tetrahedron have been achieved in Muñoz-Sola [31], Ainsworth and Demkowicz [2], and Demkowicz et al. [17], see also the references therein. Let $K$ be a triangle or a tetrahedron and let $p \ge 1$. Let $\tau_K \in \mathcal{P}_p(K)$ be a volume datum and $r_F \in \mathcal{P}_p(F)$ a target trace; the latter is prescribed on $\mathcal{F}_K^D$, a subset of all $(d-1)$-dimensional faces of $K$, possibly empty. Importantly, $r_F$ has to be compatible, a trace on $\mathcal{F}_K^D$ of some p-degree polynomial in $K$. The combination of the above-cited trace liftings gives, see [23, Lemma A.1] (with a shift by $\tau_K$),

$$ \min_{\substack{v_p \in \mathcal{P}_p(K) \\ v_p=r_F \text{ on all } F \in \mathcal{F}_K^D}} \|\nabla(\tau_K - v_p)\|_K \lesssim \min_{\substack{v \in H^1(K) \\ v=r_F \text{ on all } F \in \mathcal{F}_K^D}} \|\nabla(\tau_K - v)\|_K, \quad (C.1) $$

where the hidden constant only depends on $\kappa_K$ of (2.1) and the space dimension $d$.

p-stable broken polynomial extensions achieve a similar result to (C.1) but on patches of elements, where, crucially, the datum $\tau_K$ is a discontinuous piecewise polynomial. For patches $\omega_a$ of elements sharing a vertex and prescribed trace boundary conditions on $\partial\omega_a$ in three space dimensions, they have been established in [23, Corollaries 3.1 and 3.7], see also [13, Corollaries 3.4 and 4.4], following [8] for $d=2$.

## C.2 Larger patches and no boundary conditions

We now extend the above results in two directions. First, we consider possibly larger patches $\omega$ than merely all elements sharing a given vertex. Second, we consider the case with no trace boundary conditions prescribed on $\partial\omega$. In our application in the proof of Lemmas 5.3 and 5.4, this corresponds to $\omega$ given by a collection of extended vertex patches $\tilde{\omega}_a$ not touching $\Gamma_D$; treatment of $\Gamma_D$ is postponed to Section C.3. Our main result is:

**Theorem C.1** (p-stable broken $H^1$ polynomial extension on larger patches and without boundary conditions). Let $\mathcal{T}_\omega$ be a simplicial mesh with the corresponding open, bounded, and Lipschitz polygon or polyhedron $\omega \subset \mathbb{R}^d$, $d=2,3$, with $\bar{\omega}$ contractible. Let $\tau_{hp} \in \mathcal{P}_p(\mathcal{T}_\omega)$ be a volume datum, a discontinuous piecewise polynomial of a uniform degree $p \ge 1$. Then

$$ \min_{v_p \in \mathcal{P}_p(\mathcal{T}_\omega) \cap H^1(\omega)} \|\nabla(\tau_{hp} - v_p)\|_{\omega} \lesssim \min_{v \in H^1(\omega)} \|\nabla(\tau_{hp} - v)\|_{\omega}, \quad (C.2) $$

where the constant hidden in $\lesssim$ only depends on the shape-regularity parameter $\kappa_{\mathcal{T}_\omega} := \max_{K \in \mathcal{T}_\omega} \kappa_K$, the ratio $h_\omega/\min_{K \in \mathcal{T}_\omega} h_K$, and the space dimension $d$.

*Proof for $d=2$. The two-dimensional case can be handled as the first case for $d=3$ below, since for $d=2$, thanks to Theorem B.2, $\mathcal{T}_\omega$ is always shellable. $\square$*

*Proof for $d=3$ when $\mathcal{T}_\omega$ is shellable as per Definition B.1.* When $\mathcal{T}_\omega$ is shellable as per Definition B.1, we proceed in extension of the arguments from [23, Section 5], see also [13, Section 6.4]. Let

$$ v^* := \arg \min_{\substack{v \in H^1(\omega) \\ (v,1)_\omega=0}} \|\nabla(\tau_{hp} - v)\|_{\omega} \quad (C.3) $$

denote the infinite-dimensional $H^1(\omega)$ minimizer from the right-hand side of (C.2); the mean value constraint is employed merely for uniqueness. We present a constructive proof of (C.2) which proceeds along the shelling of Definition B.1. On each element $K_i$, $1 \le i \le |\mathcal{T}_\omega|$, we construct a suitable minimizer $\xi_i \in \mathcal{P}_p(K_i)$ and we gradually set

$$ \xi_{hp}|_{K_i} := \xi_i. \quad (C.4) $$

We then verify that

$$ \xi_{hp} \in \mathcal{P}_p(\mathcal{T}_\omega) \cap H^1(\omega), \quad \|\nabla(\tau_{hp} - \xi_{hp})\|_{\omega} \lesssim \|\nabla(\tau_{hp} - v^*)\|_{\omega}, \quad (C.5) $$

which establishes (C.2). More precisely, on each step $1 \le i \le |\mathcal{T}_\omega|$, we will verify that

$$ \|\nabla(\tau_{hp} - \xi_i)\|_{K_i} \lesssim \|\nabla(\tau_{hp} - v^*)\|_{\omega}, \quad (C.6) $$
which yields the inequality in (C.5) up to a constant depending on the shape-regularity parameter $\kappa_{\mathcal{T}_{\omega}}$ of the mesh $\mathcal{T}_{\omega}$, the ratio $h_{\omega}/\min_{K \in \mathcal{T}_{\omega}} h_K$, and the space dimension $d$. Moreover, as $\xi_i$ will have its trace prescribed by $\xi_j$ on the previously enumerated $K_j$, $\xi_{hp}$ indeed has no jumps, $\xi_{hp} \in \mathcal{P}_p(\mathcal{T}_{\omega}) \cap H^1(\omega)$.

(i) On the first element $K_1$, let

$$ \xi_1 := \arg \min_{\substack{v_p \in \mathcal{P}_p(K_1) \\ (v_p, 1)_{K_1} = 0}} \|\nabla(\tau_{hp} - v_p)\|_{K_1}. \quad (C.7) $$

This is a well-posed problem. Actually, $\nabla\xi_1 = \nabla\tau_{hp}$ on $K_1$, so that (C.6) is trivial, but let us outline the path we will take subsequently. Since the datum $\tau_{hp}|_{K_1}$ in (C.7) is polynomial, we know from (C.1) that we can pass to the infinite-dimensional level,

$$ \|\nabla(\tau_{hp} - \xi_1)\|_{K_1} \lesssim \min_{v \in H^1(K_1)} \|\nabla(\tau_{hp} - v)\|_{K_1}. \quad (C.8) $$

Then, since the infinite-dimensional minimizer $v^*$ from (C.3) restricted to the element $K_1$, $v^*|_{K_1}$, belongs to the minimization set on the right-hand side of (C.8), i.e., $v^*|_{K_1} \in H^1(K_1)$ (please note that there are no trace conditions in (C.8)), we obtain

$$ \|\nabla(\tau_{hp} - \xi_1)\|_{K_1} \lesssim \|\nabla(\tau_{hp} - v^*)\|_{K_1}, \quad (C.9) $$

which gives (C.6) for $i=1$.

(ii) On each element $K_i$ with one face shared with some previously enumerated simplex, say $F_{i,j}$ shared with $K_j$, $j < i$, we consider

$$ \xi_i := \arg \min_{\substack{v_p \in \mathcal{P}_p(K_i) \\ v_p = \xi_{hp}|_{K_j} \text{ on } F_{i,j}}} \|\nabla(\tau_{hp} - v_p)\|_{K_i}. \quad (C.10) $$

Please note that since $j < i$ and by (C.4), $\xi_{hp}|_{K_j}$ is known. Then (C.10) is well-posed; there is in particular no compatibility condition to verify, since the trace is only imposed on one face. We now again employ (C.1). This yields

$$ \|\nabla(\tau_{hp} - \xi_i)\|_{K_i} \lesssim \min_{\substack{v \in H^1(K_i) \\ v = \xi_{hp}|_{K_j} \text{ on } F_{i,j}}} \|\nabla(\tau_{hp} - v)\|_{K_i}. \quad (C.11) $$

Unfortunately, now $v^*|_{K_i}$ does not belong to the minimization set on the right-hand side of (C.11) since there is a trace condition on the face $F_{i,j}$ imposed. The fix is, for the moment, easy. Consider the face neighbor $K_j$ and let $\mathbf{T}_{i,j}: K_j \to K_i$ be the unique affine geometric mapping that maps $K_j$ to $K_i$, leaving the face $F_{i,j}$ invariant. Now we “bring” the function $v^* - \xi_{hp}$ from $K_j$ to $K_i$, forming

$$ v := v^*|_{K_i} - (v^* - \xi_{hp})|_{K_j} \circ \mathbf{T}_{i,j}^{-1}, \quad (C.12) $$

see [23, Proof of (5.11b), (1)] for the details. On $F_{i,j}$, this removes the trace of $v^*$ and brings instead the requested $\xi_{hp}|_{K_j}$, so that $v$ from (C.12) now crucially belongs to the minimization set on the right-hand side of (C.11). Consequently, we obtain

$$ \|\nabla(\tau_{hp} - \xi_i)\|_{K_i} \lesssim \|\nabla(\tau_{hp} - v^*|_{K_i} + (v^* - \xi_{hp})|_{K_j} \circ \mathbf{T}_{i,j}^{-1})\|_{K_i}. \quad (C.13) $$

Finally, by the triangle inequality and the properties of the geometric map $\mathbf{T}_{i,j}$ (recall that we suppose shape regularity of $\mathcal{T}_{\omega}$),

$$ \begin{aligned} \|\nabla(\tau_{hp} - \xi_i)\|_{K_i} &\le \|\nabla(\tau_{hp} - v^*)\|_{K_i} + \|\nabla((v^* - \xi_{hp})|_{K_j} \circ \mathbf{T}_{i,j}^{-1})\|_{K_i} \\ &\lesssim \|\nabla(\tau_{hp} - v^*)\|_{K_i} + \|\nabla(v^* - \xi_{hp})\|_{K_j} \\ &\le \|\nabla(\tau_{hp} - v^*)\|_{K_i} + \|\nabla(\tau_{hp} - v^*)\|_{K_j} + \|\nabla(\tau_{hp} - \xi_{hp})\|_{K_j} \\ &\lesssim \|\nabla(\tau_{hp} - v^*)\|_{\omega}, \end{aligned} \quad (C.14) $$

where, in the last estimate, we have employed (C.6) in $K_j$, which has been established previously since $j < i$. Thus (C.6) is established.

(iii) On each element $K_i$ with two faces shared with some previously enumerated simplices, say $F_{i,j}$ shared with $K_j$, $j < i$, and $F_{i,k}$ shared with $K_k$, $k < i$, we consider

$$ \xi_i := \arg \min_{\substack{v_p \in \mathcal{P}_p(K_i) \\ v_p = \xi_{hp}|_{K_j} \text{ on } F_{i,j} \\ v_p = \xi_{hp}|_{K_k} \text{ on } F_{i,k}}} \|\nabla(\tau_{hp} - v_p)\|_{K_i}. \quad (C.15) $$
Again, since $j < i$ and $k < i$ and by (C.4), $\xi_{hp}|_{K_j}$ and $\xi_{hp}|_{K_k}$ are known. Since the trace is imposed on two faces, problem (C.15) is well-posed if the two data $\xi_{hp}|_{K_j}$ and $\xi_{hp}|_{K_k}$ are compatible, i.e., match along the common edge of $F_{i,j}$ and $F_{i,k}$, say $e$. This is crucially implied by Definition B.1: as in a shelling, elements are always added as neighbors by a collection of faces, all the tetrahedra sharing the edge $e$ other than $K_i$ come sooner in the enumeration and the edge $e$ lies in the interior of $\omega$, cf. Figure 4 Thus, $\xi_{hp}$ is trace-continuous on all faces sharing the edge $e$ different from $F_{i,j}$ and $F_{i,k}$ and consequently univalued on $e$. We then again employ (C.1), which now yields

$$ \|\nabla(\tau_{hp} - \xi_i)\|_{K_i} \lesssim \min_{\substack{v \in H^1(K_i) \\ v = \xi_{hp}|_{K_j} \text{ on } F_{i,j} \\ v = \xi_{hp}|_{K_k} \text{ on } F_{i,k}}} \|\nabla(\tau_{hp} - v)\|_{K_i}. \quad (C.16) $$

As above in step (ii), the continuous-level minimizer $v^*$ from (C.3) restricted to $K_i$ does not belong to the minimization set on the right-hand side of (C.16) since there are two trace conditions on the two faces $F_{i,j}$ and $F_{i,k}$ imposed. We now construct a suitable $v$ in the sprit of (C.12) but which now involves the geometric mappings from all the simplices sharing the edge $e$ except for $K_i$. This is done in a “2-folding” way which replaces $v^*$ on $F_{i,j}$ and $F_{i,k}$ by respectively $\xi_{hp}|_{K_j}$ and $\xi_{hp}|_{K_k}$; the precise formula is [23, equation (5.12)]. Existence of a two-color refinement around edges of [23, Lemma B.2] is crucial at this step. Then (C.6) is established as (C.14).

(iv) Finally, on each element $K_i$ with three faces shared with some previously enumerated simplices, say $F_{i,j}$ shared with $K_j$, $j < i$, $F_{i,k}$ shared with $K_k$, $k < i$, and $F_{i,l}$ shared with $K_l$, $l < i$, we consider

$$ \xi_i := \arg \min_{\substack{v_p \in \mathcal{P}_p(K_i) \\ v_p = \xi_{hp}|_{K_j} \text{ on } F_{i,j} \\ v_p = \xi_{hp}|_{K_k} \text{ on } F_{i,k} \\ v_p = \xi_{hp}|_{K_l} \text{ on } F_{i,l}}} \|\nabla(\tau_{hp} - v_p)\|_{K_i}. \quad (C.17) $$

Again, all $\xi_{hp}|_{K_j}$, $\xi_{hp}|_{K_k}$, and $\xi_{hp}|_{K_l}$ are known at this stage. As above, the well-posedness of (C.17) is a consequence of Definition B.1: for any edge $e$ common to two of the three above faces $F_{i,j}$, $F_{i,k}$, $F_{i,l}$, the face data given by $\xi_{hp}$ are compatible, i.e., match along $e$. Employing once more (C.1), we have

$$ \|\nabla(\tau_{hp} - \xi_i)\|_{K_i} \lesssim \min_{\substack{v \in H^1(K_i) \\ v = \xi_{hp}|_{K_j} \text{ on } F_{i,j} \\ v = \xi_{hp}|_{K_k} \text{ on } F_{i,k} \\ v = \xi_{hp}|_{K_l} \text{ on } F_{i,l}}} \|\nabla(\tau_{hp} - v)\|_{K_i}. \quad (C.18) $$

As above in steps (ii) and (iii), the infinite-dimensional minimizer $v^*|_{K_i}$ does not belong to the minimization set on the right-hand side of (C.18) since there are three trace conditions on the three faces $F_{i,j}$, $F_{i,k}$, and $F_{i,l}$ imposed. From Definition B.1, as the simplices are always added as neighbors by a collection of faces, all the simplices sharing the vertex $a$ common to the three faces $F_{i,j}$, $F_{i,k}$, and $F_{i,l}$ come sooner in the enumeration than $K_i$ and the vertex $a$ does not lie on the boundary of $\omega$. This enables to construct a suitable $v$ in this sprit of (C.12) but which now involves the geometric mappings from all the simplices sharing the vertex $a$ except for $K_i$. This is done in a “3-folding” way; the precise formula is [23, equation (5.14)]. Existence of a three-color refinement around vertices of [23, Lemma B.3] is crucial at this step. Then (C.6) is established similarly to (C.14). $\square$

*Proof for $d = 3$ when $\mathcal{T}_\omega$ is not shellable as per Definition B.1.* By Remark B.3, we know that situations where $\mathcal{T}_\omega$ is not shellable as per Definition B.1 exist. For the proof of (C.2) in this case, we will use Adiprasito and Benedetti [1, Theorem A] and the results from [23, 13] on vertex patches (elements sharing a given vertex) to transfer the situation to the above proof with $\mathcal{T}_\omega$ enumerated as per Definition B.1.

Suppose that $\mathcal{T}_\omega$ has no shelling as per Definition B.1. Let $\overline{\mathcal{T}}_\omega$ be the barycentric refinement of $\mathcal{T}_\omega$, i.e., the tetrahedral submesh of $\mathcal{T}_\omega$ obtained by inserting a barycenter to each edge, face, and tetrahedron in $\mathcal{T}_\omega$, where the new tetrahedra in each tetrahedron $K \in \mathcal{T}_\omega$ have the barycenter of $K$ as vertex, see Figure 5, left. From Adiprasito and Benedetti [1, Theorem A] we know that $\overline{\mathcal{T}}_\omega$ has a shelling as per Definition B.1. Consequently, from the above proof for shellable meshes, we know that

$$ \min_{v_p \in \mathcal{P}_p(\overline{\mathcal{T}}_\omega) \cap H^1(\omega)} \|\nabla(\tau_{hp} - v_p)\|_{\omega} \lesssim \min_{v \in H^1(\omega)} \|\nabla(\tau_{hp} - v)\|_{\omega}. \quad (C.19) $$

Let us denote by $\overline{v}_p^*$ the (unique, imposing $(\overline{v}_p^*, 1)_\omega = 0$) minimizer on the left-hand side of (C.19). From $\overline{v}_p^*$, piecewise polynomial with respect to the barycentric refined mesh $\overline{\mathcal{T}}_\omega$, we now construct
<figure><img src="image_7.png" /><figcaption>Figure 5: Barycentric refinement of a tetrahedron $K \in \mathcal{T}_{\omega}$ (left), vertex patch $\mathcal{T}_{a_1}$ with vertices of the barycentric refinement indicated in $K$ (middle), $K$ with 6 subtetrahedra sharing $a_1$ (right)</figcaption></figure>

$\xi_{hp} \in \mathcal{P}_p(\mathcal{T}_{\omega}) \cap H^1(\omega)$, piecewise polynomial with respect to the original mesh $\mathcal{T}_{\omega}$. We pass through all vertices $a \in \mathcal{V}$ of the original mesh $\mathcal{T}_{\omega}$ and consider the vertex patches $\mathcal{T}_a$, cf. Figure 5, middle. We show that $\xi_{hp}$ has a comparable precision to $\overline{v_p}^*$ in that

$$ \|\nabla(\tau_{hp} - \xi_{hp})\|_{\omega} \lesssim \|\nabla(\tau_{hp} - \overline{v_p}^*)\|_{\omega}. \qquad (C.20) $$

Thus, from (C.19)–(C.20), we conclude the desired result (C.2).

Let us enumerate the vertices from $\mathcal{V}$ as $a_1, \dots, a_{|\mathcal{V}|}$. There is no specific order. We define a sequence of tetrahedral meshes $\overline{T}_{\omega,i}$ of $\omega$, $0 \le i \le |\mathcal{V}|$. We set $\overline{T}_{\omega,0} := \overline{T}_{\omega}$, the barycentric refinement mesh. For all $1 \le i \le |\mathcal{V}|$, $\overline{T}_{\omega,i}$ coincides with $\overline{T}_{\omega,i-1}$ on $\omega \setminus \omega_{a_i}$ and coarsens $\overline{T}_{\omega,i-1}$ inside $\omega_{a_i}$. On the last step, $\overline{T}_{\omega,|\mathcal{V}|} = \overline{T}_{\omega}$, the original mesh. More precisely, let $i = 1$. Then the patch subdomain $\omega_{a_1}$ is formed by tetrahedra from $\mathcal{T}_{a_1}$ sharing the vertex $a_1$, see Figure 5, middle, with barycentric refinement as displayed in Figure 5, left. The mesh $\overline{T}_{\omega,1}$ is created by coarsening of $\overline{T}_{\omega,0}$ inside $\omega_{a_1}$, where each $K \in \mathcal{T}_{a_1}$ is now only refined into 6 tetrahedra having $a_1$ as vertex in place of the barycentric refinement, as indicated in Figure 5, right. Note that $\overline{T}_{\omega,1}$ is indeed a coarsening of $\overline{T}_{\omega,0}$ and keeps the triangles on those faces from $\partial\omega_{a_1}$ that do not share the vertex $a_1$ intact. For $i > 1$, we proceed similarly, always keeping the subtriangulation of $\partial\omega_{a_i}$ except faces sharing the vertex $a_i$, but coarsening $\overline{T}_{\omega,i-1}$ inside $\omega_{a_i}$. With respect to the original mesh $\mathcal{T}_{\omega}$, the arising $\overline{T}_{\omega,i}$ adds no barycenter of 1) a tetrahedron 2) a face 3) an edge from $\mathcal{T}_{\omega}$ having $a_i$ as vertex. Thus, after passing through all vertices $a_i$ from $\mathcal{T}_{\omega}$, we do recover in $\overline{T}_{\omega,|\mathcal{V}|}$ the original mesh $\mathcal{T}_{\omega}$.

We now construct a sequence of piecewise polynomials $\xi_i \in \mathcal{P}_p(\overline{T}_{\omega,i}) \cap H^1(\omega)$, $0 \le i \le |\mathcal{V}|$. We set $\xi_0 := \overline{v_p}^*$. On each step $1 \le i \le |\mathcal{V}|$, we set $\xi_i = \xi_{i-1}$ on $\omega \setminus \omega_{a_i}$ (recall that $\overline{T}_{\omega,i}$ coincides with $\overline{T}_{\omega,i-1}$ on $\omega \setminus \omega_{a_i}$) and let

$$ \xi_i|_{\omega_{a_i}} := \arg \min_{\substack{v_p \in \mathcal{P}_p(\overline{T}_{\omega,i}|_{\omega_{a_i}}) \cap H^1(\omega_{a_i}) \\ v_p = \xi_{i-1} \text{ on faces from } \partial\omega_{a_i} \text{ not sharing } a_i}} \|\nabla(\tau_{hp} - v_p)\|_{\omega_{a_i}}. \qquad (C.21) $$

Crucially, (C.21) is an energy minimization on a vertex patch $\overline{T}_{\omega,i}|_{\omega_{a_i}}$ with the volume datum $\tau_{hp}|_{\omega_{a_i}} \in \mathcal{P}_p(\overline{T}_{\omega,i}|_{\omega_{a_i}})$ (since a piecewise polynomial on the original mesh $\mathcal{T}_{\omega}$) and with the Dirichlet boundary datum $\xi_{i-1}$ from the trace of this space. Thus, using [23, Corollaries 3.1 and 3.7] or [13, Corollary 4.4], we obtain

$$ \|\nabla(\tau_{hp} - \xi_i)\|_{\omega_{a_i}} \lesssim \min_{\substack{v \in H^1(\omega_{a_i}) \\ v = \xi_{i-1} \text{ on faces from } \partial\omega_{a_i} \text{ not sharing } a_i}} \|\nabla(\tau_{hp} - v_p)\|_{\omega_{a_i}} \le \|\nabla(\tau_{hp} - \xi_{i-1})\|_{\omega_{a_i}}, $$

where the second estimate follows as $\xi_{i-1}|_{\omega_{a_i}}$ lies in $H^1(\omega_{a_i})$ and satisfies the boundary condition. We conclude by two observations: 1) $\xi_{i-1}|_{\omega_{a_i}}$ lies in $\mathcal{P}_p(\mathcal{T}_{\omega}) \cap H^1(\omega)$ defined over the original mesh $\mathcal{T}_{\omega}$, so that we can take $\xi_{hp} := \xi_{i-1}|_{\omega_{a_i}}$; 2) the above inequality gives $\|\nabla(\tau_{hp} - \xi_i)\|_{\omega} \lesssim \|\nabla(\tau_{hp} - \xi_{i-1})\|_{\omega}$, which yields (C.20) since $|\mathcal{V}|$, the number of vertices in $\mathcal{T}_{\omega}$, is bounded by $\kappa_{\mathcal{T}_{\omega}}$ and the ratio $h_{\omega}/\min_{K \in \mathcal{T}_{\omega}} h_K$. $\square$
### C.3 Extended patches and boundary conditions

We now finally formulate the result precisely in the form needed in the proof of Lemmas 5.3 and 5.4.

**Corollary C.2** (p-stable broken $H^1$ polynomial extension on extended patches $\hat{T}_K$). Let Assumption 3.1 hold. Let $\tau_{hp} \in \mathcal{P}_p(\hat{T}_K)$ be a volume datum, a discontinuous piecewise polynomial of a uniform polynomial degree $p \ge 1$. Then

$$ \min_{v_p \in \mathcal{P}_p(\hat{T}_K) \cap H_{0,D}^1(\hat{\omega}_K)} \|\nabla(\tau_{hp} - v_p)\|_{\hat{\omega}_K} \lesssim \min_{v \in H_{0,D}^1(\hat{\omega}_K)} \|\nabla(\tau_{hp} - v)\|_{\hat{\omega}_K}, \quad (C.22) $$

where the constant hidden in $\lesssim$ only depends on the shape-regularity parameter $\kappa_{\hat{T}_K}$ of the mesh $\hat{T}_K$ given by $\kappa_{\hat{T}_K} := \max_{L \in \hat{T}_K} \kappa_L$ and the space dimension $d$.

*Proof.* Let $\hat{\omega}_K$ have no face on the Dirichlet boundary $\Gamma_D$, $(\partial\hat{\omega}_K \cap \Gamma_D)^\circ = \emptyset$. Then (C.22) follows by Theorem C.1; note that the ratio $h_{\hat{\omega}_K}/\min_{L \in \hat{T}_K} h_L$ for an extended patch $\hat{T}_K$ only depends on the shape-regularity parameter $\kappa_{\hat{T}_K}$ and space dimension $d$. In the case $(\partial\hat{\omega}_K \cap \Gamma_D)^\circ \neq \emptyset$ but when the boundary of $\hat{\omega}_K$ does not coincide with the Dirichlet boundary $\Gamma_D$, one can proceed following [23, Section 7], [13, Section 7], and the proof of Theorem C.1, case $d=3$ when $\mathcal{T}_\omega$ has no shelling as per Definition B.1. Here, one designs a sequence of mappings where one can deduce the validity of (C.22) in the case $(\partial\hat{\omega}_K \cap \Gamma_D)^\circ \neq \emptyset$ from the case $(\partial\hat{\omega}_K \cap \Gamma_D)^\circ = \emptyset$. $\square$

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