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Uniform Poincaré inequalities for the discrete de Rham complex of differential forms

Daniele Di Pietro, Jérôme Droniou, Marien-Lorenzo Hanot, Silvano Pitassi

TL;DR

The paper proves a uniform-in-mesh Poincaré inequality for the discrete de Rham (DDR) complex on polytopal meshes in arbitrary dimension and topology. It reduces the problem to a Poincaré inequality for cochains on the mesh and constructs a graded, cochain-preserving map that lifts polytopal cochains to simplicial cochains, via Whitney and de Rham correspondences. The main result provides stability bounds for discrete forms in terms of their discrete exterior derivatives, uniformly with respect to mesh size, and bridges to standard Poincaré inequalities on kernels and orthogonal complements. This yields foundational stability for Mimetic Finite Differences, Compatible Discrete Operators, and Discrete Geometric Approaches on generic polytopal domains.

Abstract

In this paper we prove discrete Poincaré inequalities that are uniform in the mesh size for the discrete de Rham complex of differential forms developed in [Bonaldi, Di Pietro, Droniou, and Hu, An exterior calculus framework for polytopal methods, J. Eur. Math. Soc., to appear, arXiv preprint 2303.11093]. We unify the underlying ideas behind the Poincaré inequalities for all differential operators in the sequence, extending the known inequalities for the gradient, curl, and divergence in three-dimensions to polytopal domains of arbitrary dimension and general topology. A key step in the proof involves deriving specific Poincaré inequalities for the cochain complex supported on the polytopal mesh. These inequalities are of independent interest, as they are useful, for instance, in establishing the existence and stability, on domains of generic topology, of solutions of schemes based on Mimetic Finite Differences, Compatible Discrete Operators or Discrete Geometric Approach.

Uniform Poincaré inequalities for the discrete de Rham complex of differential forms

TL;DR

The paper proves a uniform-in-mesh Poincaré inequality for the discrete de Rham (DDR) complex on polytopal meshes in arbitrary dimension and topology. It reduces the problem to a Poincaré inequality for cochains on the mesh and constructs a graded, cochain-preserving map that lifts polytopal cochains to simplicial cochains, via Whitney and de Rham correspondences. The main result provides stability bounds for discrete forms in terms of their discrete exterior derivatives, uniformly with respect to mesh size, and bridges to standard Poincaré inequalities on kernels and orthogonal complements. This yields foundational stability for Mimetic Finite Differences, Compatible Discrete Operators, and Discrete Geometric Approaches on generic polytopal domains.

Abstract

In this paper we prove discrete Poincaré inequalities that are uniform in the mesh size for the discrete de Rham complex of differential forms developed in [Bonaldi, Di Pietro, Droniou, and Hu, An exterior calculus framework for polytopal methods, J. Eur. Math. Soc., to appear, arXiv preprint 2303.11093]. We unify the underlying ideas behind the Poincaré inequalities for all differential operators in the sequence, extending the known inequalities for the gradient, curl, and divergence in three-dimensions to polytopal domains of arbitrary dimension and general topology. A key step in the proof involves deriving specific Poincaré inequalities for the cochain complex supported on the polytopal mesh. These inequalities are of independent interest, as they are useful, for instance, in establishing the existence and stability, on domains of generic topology, of solutions of schemes based on Mimetic Finite Differences, Compatible Discrete Operators or Discrete Geometric Approach.

Paper Structure

This paper contains 28 sections, 19 theorems, 138 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main results
  3. Setting
  4. Polytopal mesh
  5. Discrete differential forms
  6. Local spaces of forms with polynomial coefficients
  7. The Discrete De Rham construction
  8. Spaces and interpolators
  9. Local construction
  10. Component norms
  11. Poincaré inequalities for the Discrete de Rham complex
  12. A Poincaré inequality for cochains
  13. Setting
  14. Chain complex
  15. Cochain complex
  16. ...and 13 more sections

Key Result

Theorem 4

Let $k \in \{0,\ldots,n-1\}$ and let $\ul\omega_h\in \underline{X}_{r,h}^{k}$. Then, there exists $\ul\tau_h \in \underline{X}_{r,h}^{k}$ such that $\ul\mathrm{d}^k_{r,h} \ul\tau_h = \ul\mathrm{d}^k_{r,h} \ul\omega_h$ and Equivalently,

Figures (2)

  • Figure 1: (a) A simplicial subdivision $\mathcal{S}_h(f)$ of a pyramidal $3$-cell $f$ with a quadrilateral base $f'$; $\mathcal{S}_h(f)$ is made of four $3$-simplices so that $\Delta_{3}(\mathcal{S}_h(f)) = \{S_1,S_2, S_3, S_4\}$. (b) Case $(k,d)=(2,3)$ ($2$-simplices contained in the pyramid). A exploded view of the set $\mathcal{F}_2 = \{F_1, F_2, F_3\}$ of $2$-simplices associated with the basis $\mathcal{B}_2$ computed with Algorithm \ref{['acyclic_algo']} in Appendix \ref{['section:algorithm']}. In red and blue, the $2$-cycle $z_1 \in \mathcal{B}_2$ satisfying $\langle {F_i} \, , {z_1} \rangle = \delta_{i1}$ for $i \in \{1,2,3\}$; the $3$-chain $w_1$ such that $\partial_{3} w_1 = z_1$ is $w_1 = S_1$. In blue, the restriction $z_1|_{\partial f} \in C_{2}(\mathcal{S}_h(f))$ of $z_1$ to $\partial f$. (c) Case $(k,d) = (1,2)$ (left, corresponding to $1$-simplices contained in a polygonal $2$-cell) and $(k,d) = (1,3)$ (right, corresponding to $1$-simplices contained in the pyramid). On the left, $d=2$, and the set $\mathcal{F}_1 = \{F_1, F_2, F_3\}$ of 1-simplices associated with the basis $\mathcal{B}_1$ computed with Algorithm \ref{['acyclic_algo']}. In red and blue, the $1$-cycle $z_1 \in \mathcal{B}_1$ satisfying $\langle {F_i} \, , {z_1} \rangle = \delta_{i1}$ for $i \in \{1,2,3\}$. In blue, the restriction $z_1|_{\partial f} \in C_{1}(\mathcal{S}_h(f))$ of $z_1$ to $\partial f$, and in light grey the $2$-chain $w_1$ such that $\partial_{2} w_1 = z_1$. On the right, $d=3$, and the set $\mathcal{F}_1 = \{F_1\}$. In red, blue, and light grey the corresponding $1$-chains $z_1$, $z_1|_{\partial f}$, and $2$-chain $w_1$ (supported by two $2$-simplices). (d) Case $(k,d) = (0,2)$. The set $\mathcal{F}_0 = \{F_1\}$ associated with the basis $\mathcal{B}_0$ computed with Algorithm \ref{['acyclic_algo']}, and the $0$-cell $F^* \in \Delta_{0}(\mathcal{S}_h(\partial f'))$. By the substitution in \ref{['eq:substitution.b']}, the $0$-chain $z_1 \in \mathcal{B}_0$ is $z_1 \coloneqq F_1 - F^*$, and its restriction $z_1|_{\partial f'}$ to $\partial f'$ is the $0$-chain $z_1|_{\partial f'} = -F^*$. The $1$-chain $w_1$ such that $\partial_{1}w_1 = z_1$ is then simply the $1$-simplex in $\mathcal{S}_h(f')$ connecting $F_1$ and $F^*$.
  • Figure 2: Illustration of the $2$-chain $w_{p+1}$ (light grey) and $1$-chain $z_{p+1}$ (red) for a simplicial subdivision $\mathcal{S}_h(f)$ of a quadrilateral $2$-cell $f$. Note how the orientation of the $2$-simplex $F$ is the same as that of the $2$-cell $f$, as well as how $F$ and $f$ induce the same orientation on the unique $1$-cell $f' = F'$ (also, $1$-simplex in this case) which lies in their intersection $\partial F \cap \partial f = \{f'\}=\{F'\}$ so that $\epsilon_{FF'} = \epsilon_{f\!f'}$.

Theorems & Definitions (44)

  • Remark 1: Integral formulas
  • Remark 2: Equivalent component norms
  • Remark 3: Properties of the potential reconstruction
  • Theorem 4: Poincaré inequality
  • proof
  • Corollary 5: Poincaré inequality on orthogonal complements
  • proof
  • Lemma 5: Poincaré inequality for cochains
  • proof
  • Remark 6: Interpretation of Lemma \ref{['lemma:topological.bound']} in the context of low-order compatible methods
  • ...and 34 more