n-dimensional triangulated manifolds.

# Acknowledgments

MWL appreciates helpful remarks by Abdellatif Aitelhad, Martin Costabel, John M. Lee, and Günther M. Ziegler.

# A Outline of an alternative estimate

We explore a family of alternative estimates, based on a variation of the results of Section 8 and Section 9. Though we consider these ideas to be of interest, the resulting estimates for Poincaré-Friedrichs constants are not competitive in practice.

We begin with an analogue of Proposition 8.3. Domains with shellable triangulations are homeomorphic to a ball and hence contractible. Inspired by that observation, we show that shellable triangulations can be continuously retracted into a single simplex via a sequence of bi-Lipschitz transformations. At each step, this bi-Lipschitz transformation is defined as contracting a simplex T into a local star around one of the subsimplices of T.

**Proposition A.1.** Let $\mathcal{T}$ be a triangulation of an $n$-dimensional domain. Let $S \in \mathcal{T}$ be an inner simplex of dimension $l < n$, let $T \in \text{st}_{\mathcal{T}}(S)$ be of dimension $n$, and let

$$U := |\text{st}_{\mathcal{T}}(S)| \setminus \bar{T}, \quad \Gamma_2 := \overline{\partial U \setminus \partial \bar{T}}.$$

The following holds, where the constants on the right-hand sides are as stated in the proof.

There exists a bi-Lipschitz piecewise affine mapping

$$\Xi_2 : |\text{st}_{\mathcal{T}}(S)| \to \Xi_2(|\text{st}_{\mathcal{T}}(S)|) \subseteq U$$

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

$$\begin{gathered} \sigma_1 \le C_{7,n,l}(\mathcal{T}), \quad \sigma_2, \dots, \sigma_n \le C'_{7,n,l}(\mathcal{T}), \quad |\det \mathbf{J}\Xi_2| \le C^{\det}_{7,n,l}(\mathcal{T}), \\ \sigma^{-1}_n \le C_{8,n,l}(\mathcal{T}), \quad \sigma^{-1}_1, \dots, \sigma^{-1}_{n-1} \le C'_{8,n,l}(\mathcal{T}), \quad |\det \mathbf{J}\Xi_2^{-1}| \le C^{\det}_{8,n,l}(\mathcal{T}). \end{gathered}$$

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

$$\sigma_1 \cdots \sigma_k |\det \mathbf{J}\Xi_2|^{-\frac{1}{p}} \le C_{7,n,k,l,p}(\mathcal{T}), \quad \sigma_n^{-1} \cdots \sigma_{n-k+1}^{-1} |\det \mathbf{J}\Xi_2|^{\frac{1}{p}} \le C_{8,n,k,l,p}(\mathcal{T}).$$

*Proof.* This proof is to be read as a continuation of the proof of Proposition 8.3, with the same notation and definitions.

Notice that $\partial|\text{st}_{\mathcal{T}}(S)| \setminus \Gamma_2$ is the union of exactly those faces of $T$ that contain $S'$.

We introduce another mapping $\Phi : |\mathcal{K}^*| \to |\mathcal{K}^c|$ as follows. Consider any $n$-simplex $K \in \mathcal{K}$ and let $K^c \in \mathcal{K}^c$ be its counterpart as defined earlier in the proof. We construct a bi-Lipschitz mapping

$$\Phi_K : K \cup K^c \to K^c.$$

The construction will be such that the union of $\Phi_K$ for all $n$-simplices $K \in \mathcal{K}$ will define the desired bi-Lipschitz mapping $\Phi : |\mathcal{K}^*| \to |\mathcal{K}^c|$, and $\Phi$ will be the identity along $\partial|\mathcal{K}^*| \setminus \partial|\mathcal{K}|$.

Once again, $h_z$ denotes the height of $z_{S'}$ within $K$. Here, we let $Q \subseteq K \cap K^c$ be the subsimplex that is complementary to the line segment from the origin to $z_{S'}$ in $K$. Equivalently, $Q$ is complementary to the line segment from the origin to $y$ in $K^c$, and $Q$ is complementary to the origin within the face $K \cap K^c$. From the definition of simplices, we now conclude that any $x \in K \cup K^c$ has a unique representation

$$x = \lambda z_{S'} + \mu x_Q, \quad \lambda \in [-\rho, 1], \quad \mu \in [0, 1], \quad |\lambda| + \mu \le 1, \quad x_Q \in Q.$$

Since $\mu x_Q$ lies in the hyperplane spanned by the origin and $Q$, on which $h_z$ stands orthogonally, we have

$$\lambda = \lambda(x) := \frac{\langle h_z, x \rangle}{\langle h_z, z_{S'} \rangle}.$$
Based on that observation, we define

$$
\begin{align*}
\Phi_K(x) &:= \mu x_Q + \frac{\rho}{\rho+1} (\lambda(x) - 1) z_{S'} \\
&= \mu x_Q + \frac{\rho}{\rho+1} \lambda(x) z_{S'} - \frac{\rho}{\rho+1} z_{S'} \\
&= x - \frac{\rho+1}{\rho+1} \lambda(x) z_{S'} + \frac{\rho}{\rho+1} \lambda(x) z_{S'} - \frac{\rho}{\rho+1} z_{S'} \\
&= x - \frac{1}{\rho+1} \lambda(x) z_{S'} - \frac{\rho}{\rho+1} z_{S'} \\
&= x - \frac{1}{\rho+1} \frac{\langle h_z, x \rangle}{\langle h_z, z_{S'} \rangle} z_{S'} - \frac{\rho}{\rho+1} z_{S'}.
\end{align*}
$$

We readily verify that this transformation is a bi-Lipschitz mapping from $K \cup K^c$ onto $K^c$ that satisfies the desired mapping properties. It remains to analyze its Jacobian and get explicit estimates for its singular values.

We once more introduce an orthogonal decomposition $h_z + b_z = z_{S'}$, where $b_z \in \mathbb{R}^n$. With that,

$$
\begin{align*}
\mathbf{J} \Phi_K(x) &= \mathrm{Id} - \frac{1}{1+\rho} \frac{z_{S'} \otimes h_z^t}{\langle h_z, z_{S'} \rangle} \\
&= \mathrm{Id} - \frac{1}{1+\rho} \frac{z_{S'} \otimes h_z^t}{\langle h_z, h_z \rangle} \\
&= \mathrm{Id} - \frac{1}{1+\rho} \frac{h_z \otimes h_z^t}{\langle h_z, h_z \rangle} - \frac{1}{1+\rho} \frac{b_z \otimes h_z^t}{\langle h_z, h_z \rangle} \\
&= \mathrm{Id} - \frac{1}{1+\rho} \hat{h}_z \otimes \hat{h}_z^t - \frac{\|b_z\|/\|h_z\|}{1+\rho} \hat{b}_z \otimes \hat{h}_z^t.
\end{align*}
$$

The Jacobian acts as the identity over the orthogonal complement of the span of $h_z$ and $z_{S'}$. We write $\beta$ for the angle between $h_z$ and $z_{S'}$. Hence, $\tan(\beta) = \|b_z\|/\|h_z\|$. Direct computation shows

$$
\mathbf{J} \Phi_K(x) \cdot \hat{b}_z = \hat{b}_z, \quad \mathbf{J} \Phi_K(x) \cdot \hat{h}_z = \frac{\rho}{1+\rho} \hat{h}_z - \frac{\|b_z\|/\|h_z\|}{1+\rho} \hat{b}_z. \tag{65}
$$

It remains to study the singular values of the matrix

$$
M_{\Phi,K} = \begin{pmatrix} \frac{\rho}{1+\rho} & 0 \\ -\frac{\tan(\beta)}{1+\rho} & 1 \end{pmatrix}.
$$

Computing the eigenvalues of the symmetric matrix $M_{\Phi,K}^t M_{\Phi,K}$, we obtain the singular values

$$
\sigma_{\max}(\Phi, K) = \frac{1}{2(1+\rho)} \sqrt{(2\rho+1)^2 + \tan(\beta)^2} + \frac{1}{2(1+\rho)} \sqrt{1 + \tan(\beta)^2},
$$

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

These are monotonically increasing from 1 and decreasing from 0.5, respectively, in tan(β). Hence, these
are also the maximal and minimal singular values of the Jacobian **JΦ**<sub>K</sub>, the remaining singular values
being equal to 1. Notice that

$$
\det \mathbf{J} \Phi = \sigma_{\max}(\Phi, K) \sigma_{\min}(\Phi, K) = \frac{\rho(1+\rho)}{(1+\rho)^2} = \frac{\rho}{1+\rho}.
$$

We now recall that the height of $h_z$ in $K \in \mathcal{K}$ equals $(\ell+1)^{-1}$ multiplied with the height of some vertex of $S$ within $T$. Similar as above, we use the upper bound

$$
\tan(\beta) = \frac{\|b_z\|}{\|h_z\|} \le (\ell + 1)\kappa_A(T).
$$
We conclude that the singular values of the Jacobian of $\Phi$ at almost every $x$ satisfy

$$ \sigma_1(\Phi, x) \le \frac{1}{2(1+\rho)} \sqrt{(2\rho+1)^2 + (\ell+1)^2 \kappa_A(T)^2} + \frac{1}{2(1+\rho)} \sqrt{1 + (\ell+1)^2 \kappa_A(T)^2}, \quad (66) $$

$$ \sigma_2(\Phi, x) = \dots = \sigma_{n-1}(\Phi, x) = 1, \quad (67) $$

$$ \sigma_n(\Phi, x)^{-1} = \frac{\sigma_1(\Phi, x)}{|\det \mathbf{J}\Phi|} \le \frac{1}{2\rho} \sqrt{(2\rho+1)^2 + (\ell+1)^2 \kappa_A(T)^2} + \frac{1}{2\rho} \sqrt{1 + (\ell+1)^2 \kappa_A(T)^2}. \quad (68) $$

This finishes the discussion of the transformation $\Phi$. Finally, we obtain $\Xi_2$ by extending $\Phi$ to the entire patch as the identity. □

**Remark A.2.** We notice that, similar to the mapping $\Xi_1$, the mapping $\Xi_2$ is not only bi-Lipschitz onto its image but also piecewise affine with respect to some essentially non-overlapping decomposition of its respective domains into simplices.

We are now ready to describe alternative estimates for potential operators that rely on Proposition A.1. Let $\mathcal{T}$ be a shellable $n$-dimensional triangulation, and let the domain $\Omega \subseteq \mathbb{R}^n$ be the interior of the underlying set of $\mathcal{T}$.

Let $T_0, T_1, \dots, T_M$ be a shelling of $\mathcal{T}$. For each simplex $T_1, \dots, T_M$ there exists a subsimplex $S_1 \subseteq T_1, \dots, S_M \subseteq T_M$ such that adding $T_m$ to the intermediate triangulation ($1 \le m \le M$) completes the local star $\mathrm{st}_{\mathcal{T}}(S_m)$ around $S_m$. Proposition A.1 shows that for each index, there exists a bi-Lipschitz contraction which maps $\mathrm{st}_{\mathcal{T}}(S)$ onto $\mathrm{st}_{\mathcal{T}}(S_m) \setminus S_m$ and is the identity on all other simplices joined so far.

Evidently, the composition of these bi-Lipschitz contractions contracts $\Omega$ onto the interior of $T_0$. Every shellable triangulation can be transformed into a single simplex along a sequence of local bi-Lipschitz (in fact, piecewise linear) deformations, and their bi-Lipschitz constants are controlled by the shape regularity of the triangulation.

Suppose now we want to find a potential for some differential form of degree $k+1$ over the triangulated domain $\Omega$. Using successive pullbacks along the local contractions, the original potential problem over the domain is reduced to a potential problem for some right-hand side over the first simplex of the shelling. The potential over that first simplex, a $k$-form, is then extended along successive reverse pullbacks to the whole domain. Since pullbacks commute with the exterior derivative, we thus obtain a solution to the original potential problem.

Potential estimates over the first single simplex, which is convex, are well-established. Together with norm estimates for the pullback of forms (Proposition 5.3), we derive estimates for the solution. We summarize this in the following theorem.

**Theorem A.3.** Let $\mathcal{T}$ be a shellable $n$-dimensional triangulation, and let the domain $\Omega \subseteq \mathbb{R}^n$ be the interior of the underlying set of $\mathcal{T}$. Let $T_0, T_1, \dots, T_M$ be a shelling of $\mathcal{T}$ and let $0 \le \ell_m < n$, for $1 \le m \le M$, be such that $T_m$ has $n - \ell_m$ faces in common with the previous simplices. Then for any $u \in W^p \Lambda^k(\Omega)$, where $1 \le p \le \infty$, there exists $w \in W^p \Lambda^k(\Omega)$ with $dw = du$ such that

$$ \|w\|_{L^p(\Omega)} \le \left( \prod_{m=1}^{M} C_{8,n,k+1,\ell_m,p}(\mathcal{T}) C_{7,n,k,\ell_m,p}(\mathcal{T}) \right) C_{\mathrm{PF},T_0,k,p} \|du\|_{L^p(\Omega)}. $$

However, in our numerical examples (see Section 10), the estimate performs significantly worse than the estimate derived in Theorem 9.3, which is why we do not recommend this method for practical computations.

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