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Commuting quasi-interpolators and Maxwell compactness for a polytopal de Rham complex

Théophile Chaumont-Frelet, Jérôme Droniou, Simon Lemaire

TL;DR

The paper addresses the challenge of establishing Maxwell-type compactness for polytopal Discrete De Rham ($\mathrm{DDR}$) complexes and designing commuting quasi-interpolators that map minimal-regularity de Rham spaces onto DDR spaces. It develops a full framework for the DDR on generic polygonal/polyhedral meshes, including primal and adjoint consistency results, and uses liftings to connect the discrete and continuous complexes. The main contributions are the first Maxwell compactness theorems for a polytopal de Rham complex and a construction of commuting quasi-interpolators with robust stability and consistency properties, paving the way for convergence proofs of DDR schemes under minimal regularity and mixed boundary conditions. The results provide a solid theoretical foundation for reliable, parameter-robust DDR discretizations of electromagnetic and related PDEs on general meshes, with potential applicability to other polytopal or virtual element frameworks.

Abstract

We establish Maxwell compactness results for the Discrete De Rham (DDR) polytopal complex: sequences in this polytopal complex with bounded discrete $\boldsymbol{H}(\mathbf{curl})$ (resp. discrete $\boldsymbol{H}(\mathrm{div})$) norm and orthogonal to discrete gradients (resp. discrete curls) have $L^2$-relatively compact potential reconstructions. The proof of these results hinges on the design of novel quasi-interpolators, that map the minimal-regularity de Rham spaces onto the discrete DDR spaces and form a commuting diagram. A full set of (primal and adjoint) consistency properties is established for these quasi-interpolators, which paves the way to convergence proofs, under minimal-regularity assumptions, of DDR schemes for partial differential equations based on the de Rham complex. Our analysis is performed with generic mixed boundary conditions, also covering the cases of no boundary conditions or fully homogeneous boundary conditions, and leverages recently introduced liftings from the DDR complex to the continuous de Rham complex.

Commuting quasi-interpolators and Maxwell compactness for a polytopal de Rham complex

TL;DR

The paper addresses the challenge of establishing Maxwell-type compactness for polytopal Discrete De Rham () complexes and designing commuting quasi-interpolators that map minimal-regularity de Rham spaces onto DDR spaces. It develops a full framework for the DDR on generic polygonal/polyhedral meshes, including primal and adjoint consistency results, and uses liftings to connect the discrete and continuous complexes. The main contributions are the first Maxwell compactness theorems for a polytopal de Rham complex and a construction of commuting quasi-interpolators with robust stability and consistency properties, paving the way for convergence proofs of DDR schemes under minimal regularity and mixed boundary conditions. The results provide a solid theoretical foundation for reliable, parameter-robust DDR discretizations of electromagnetic and related PDEs on general meshes, with potential applicability to other polytopal or virtual element frameworks.

Abstract

We establish Maxwell compactness results for the Discrete De Rham (DDR) polytopal complex: sequences in this polytopal complex with bounded discrete (resp. discrete ) norm and orthogonal to discrete gradients (resp. discrete curls) have -relatively compact potential reconstructions. The proof of these results hinges on the design of novel quasi-interpolators, that map the minimal-regularity de Rham spaces onto the discrete DDR spaces and form a commuting diagram. A full set of (primal and adjoint) consistency properties is established for these quasi-interpolators, which paves the way to convergence proofs, under minimal-regularity assumptions, of DDR schemes for partial differential equations based on the de Rham complex. Our analysis is performed with generic mixed boundary conditions, also covering the cases of no boundary conditions or fully homogeneous boundary conditions, and leverages recently introduced liftings from the DDR complex to the continuous de Rham complex.
Paper Structure (19 sections, 10 theorems, 118 equations)

This paper contains 19 sections, 10 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. The Discrete De Rham complex
  3. Mesh
  4. Functional spaces
  5. Sobolev spaces
  6. Local polynomial and finite element spaces
  7. The DDR complex
  8. Components and $L^2$-like norms
  9. Compactness results for the DDR complex
  10. Quasi-interpolators for the DDR complex
  11. Design and cochain property
  12. Consistency properties
  13. Analysis of the quasi-interpolators
  14. Proof of the bounded cochain map property (Theorem \ref{['thm:qI.cochain']})
  15. Consistency properties of the quasi-interpolators
  16. ...and 4 more sections

Key Result

Theorem 2

Consider $\mu$ that satisfies Assumption assum:physical.parameter. Let $(\underline{\boldsymbol{v}}_{h})_{h\in\mathcal{H}}$ be such that $\underline{\boldsymbol{v}}_{h}\in\underline{X}_{\mathop{\mathrm{\mathbf{curl}}}\nolimits,\Gamma}^{k}(\mathcal{T}_h)$ for each $h\in\mathcal{H}$, and Then, there exists $\boldsymbol{v}\in\boldsymbol{H}_{\Gamma}(\mathop{\mathrm{\mathbf{curl}}}\nolimits,\Omega)\ca

Theorems & Definitions (29)

  • Theorem 2: Maxwell compactness (I)
  • proof
  • Theorem 3: Maxwell compactness (II)
  • proof
  • Theorem 4: Discrete Rellich theorem
  • proof
  • Remark 5: $L^2$-bound assumptions
  • Remark 6: Generic notation and local interpolator
  • Theorem 7: Bounded cochain map
  • proof
  • ...and 19 more