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Uniform Poincaré inequalities for the Discrete de Rham complex on general domains

Daniele A. Di Pietro, Marien-Lorenzo Hanot

TL;DR

This work develops a unified framework of discrete Poincaré inequalities for the 3D Discrete de Rham (DDR) complex on general polyhedral domains, extending stability results to domains with nontrivial topology. By introducing mimetic Poincaré inequalities and a tetrahedral submesh, the authors derive gradient, curl, and divergence bounds that tie $L^2$-type norms of polynomial vectors to discrete operators, and generalize curl results to domains with arbitrary second Betti numbers. The main contributions include continuous-inverse arguments for the discrete operators and a stability proof for a DDR magnetostatics scheme on topologically complex domains, augmented by a well-structured DDR complex and compatible norms. The results inform robust, topology-aware discretizations in computational electromagnetism and more broadly for polytopal exterior calculus methods. Overall, the paper provides a rigorous, topology-sensitive foundation for stable DDR-based approximations on general domains.

Abstract

In this paper we prove Poincaré inequalities for the Discrete de Rham (DDR) sequence on a general connected polyhedral domain $Ω$ of $\mathbb{R}^3$. We unify the ideas behind the inequalities for all three operators in the sequence, deriving new proofs for the Poincaré inequalities for the gradient and the divergence, and extending the available Poincaré inequality for the curl to domains with arbitrary second Betti numbers. A key preliminary step consists in deriving "mimetic" Poincaré inequalities giving the existence and stability of the solutions to topological balance problems useful in general discrete geometric settings. As an example of application, we study the stability of a novel DDR scheme for the magnetostatics problem on domains with general topology.

Uniform Poincaré inequalities for the Discrete de Rham complex on general domains

TL;DR

This work develops a unified framework of discrete Poincaré inequalities for the 3D Discrete de Rham (DDR) complex on general polyhedral domains, extending stability results to domains with nontrivial topology. By introducing mimetic Poincaré inequalities and a tetrahedral submesh, the authors derive gradient, curl, and divergence bounds that tie -type norms of polynomial vectors to discrete operators, and generalize curl results to domains with arbitrary second Betti numbers. The main contributions include continuous-inverse arguments for the discrete operators and a stability proof for a DDR magnetostatics scheme on topologically complex domains, augmented by a well-structured DDR complex and compatible norms. The results inform robust, topology-aware discretizations in computational electromagnetism and more broadly for polytopal exterior calculus methods. Overall, the paper provides a rigorous, topology-sensitive foundation for stable DDR-based approximations on general domains.

Abstract

In this paper we prove Poincaré inequalities for the Discrete de Rham (DDR) sequence on a general connected polyhedral domain of . We unify the ideas behind the inequalities for all three operators in the sequence, deriving new proofs for the Poincaré inequalities for the gradient and the divergence, and extending the available Poincaré inequality for the curl to domains with arbitrary second Betti numbers. A key preliminary step consists in deriving "mimetic" Poincaré inequalities giving the existence and stability of the solutions to topological balance problems useful in general discrete geometric settings. As an example of application, we study the stability of a novel DDR scheme for the magnetostatics problem on domains with general topology.
Paper Structure (22 sections, 14 theorems, 130 equations)

This paper contains 22 sections, 14 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Discrete de Rham construction
  3. Domain and mesh
  4. Polynomial spaces and $L^2$-orthogonal projectors
  5. DDR spaces
  6. Local vector calculus operators and potentials
  7. Gradient
  8. Curl
  9. Divergence
  10. DDR complex
  11. Component norms
  12. Main results
  13. Mimetic Poincaré inequalities
  14. Mimetic Poincaré inequality for collections of vertex values
  15. Mimetic Poincaré inequality for collections of edge values
  16. ...and 7 more sections

Key Result

Theorem 1

For all $\underline{p}_h \in \underline{X}_{\mathop{\mathrm{\bf grad}}\nolimits,h}^k$, it holds with hidden constant only depending on $\Omega$, the mesh regularity parameter, and $k$.

Theorems & Definitions (29)

  • Theorem 1: Poincaré inequality for the gradient
  • proof
  • Theorem 2: Poincaré inequality for the curl
  • proof
  • Theorem 3: Poincaré inequality for the divergence
  • proof
  • Theorem 4: Mimetic Poincaré inequality for collections of vertex values
  • proof
  • Lemma 5: Condition on the cohomology
  • proof
  • ...and 19 more