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Conforming lifting and adjoint consistency for the Discrete de Rham complex of differential forms

Daniele A. Di Pietro, Jérôme Droniou, Silvano Pitassi

TL;DR

The paper addresses the challenge of non-conforming yet compatible discrete de Rham complexes on polytopal meshes by constructing explicit conforming liftings that are right inverses of the DDR interpolators. It proves adjoint-consistency estimates for the discrete exterior derivative using these liftings, within a unified polytopal exterior calculus framework. The lifting is built via a hierarchical, cell-wise local construction in trimmed polynomial spaces, ensuring piecewise polynomiality, trace compatibility, and stability. These results pave the way for rigorous analysis of DDR schemes and potential extensions to discrete functional-analytic inequalities and compactness results for compatible polytopal methods.

Abstract

Discrete de Rham (DDR) methods provide non-conforming but compatible approximations of the continuous de Rham complex on general polytopal meshes. Owing to the non-conformity, several challenges arise in the analysis of these methods. In this work, we design conforming liftings on the DDR spaces, that are right-inverse of the interpolators and can be used to solve some of these challenges. We illustrate this by tackling the question of the global integration-by-part formula. By non-conformity of the discrete complex, this formula involves a residual -- which can be interpreted as a consistency error on the adjoint of the discrete exterior derivative -- on which we obtain, using the conforming lifting, an optimal bound in terms of the mesh size. Our analysis is carried out in the polytopal exterior calculus framework, which allows for unified proofs for all the spaces and operators in the DDR complex. Moreover, the liftings are explicitly constructed in finite element spaces on a simplicial submesh of the underlying polytopal mesh, which gives more control on the resulting functions (e.g., discrete trace and inverse inequalities).

Conforming lifting and adjoint consistency for the Discrete de Rham complex of differential forms

TL;DR

The paper addresses the challenge of non-conforming yet compatible discrete de Rham complexes on polytopal meshes by constructing explicit conforming liftings that are right inverses of the DDR interpolators. It proves adjoint-consistency estimates for the discrete exterior derivative using these liftings, within a unified polytopal exterior calculus framework. The lifting is built via a hierarchical, cell-wise local construction in trimmed polynomial spaces, ensuring piecewise polynomiality, trace compatibility, and stability. These results pave the way for rigorous analysis of DDR schemes and potential extensions to discrete functional-analytic inequalities and compactness results for compatible polytopal methods.

Abstract

Discrete de Rham (DDR) methods provide non-conforming but compatible approximations of the continuous de Rham complex on general polytopal meshes. Owing to the non-conformity, several challenges arise in the analysis of these methods. In this work, we design conforming liftings on the DDR spaces, that are right-inverse of the interpolators and can be used to solve some of these challenges. We illustrate this by tackling the question of the global integration-by-part formula. By non-conformity of the discrete complex, this formula involves a residual -- which can be interpreted as a consistency error on the adjoint of the discrete exterior derivative -- on which we obtain, using the conforming lifting, an optimal bound in terms of the mesh size. Our analysis is carried out in the polytopal exterior calculus framework, which allows for unified proofs for all the spaces and operators in the DDR complex. Moreover, the liftings are explicitly constructed in finite element spaces on a simplicial submesh of the underlying polytopal mesh, which gives more control on the resulting functions (e.g., discrete trace and inverse inequalities).

Paper Structure

This paper contains 18 sections, 8 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. The Discrete de Rham complex
  3. Setting
  4. Polytopal mesh
  5. Differential forms
  6. Local polynomial forms
  7. The Discrete de Rham construction
  8. Spaces, interpolators and discrete operators
  9. Norms and discrete inner products
  10. Main results
  11. Conforming lifting
  12. Consistency properties
  13. Proof of the consistency properties
  14. Construction of the conforming lifting
  15. Cochain complexes: preliminaries and local problem
  16. ...and 3 more sections

Key Result

Theorem 3

Let $\mathcal{M}_h$ be a polytopal mesh, $r\ge 0$ be a polynomial degree, and $k\in\{0,\ldots,n\}$ be a form degree. Then, there exists a linear lifting ${\mathcal{L}}^{k}_{r,h}:\underline{X}_{r,h}^{k}\to H\Lambda^{k}(\Omega)$ such that, for all $\ul{\omega}_h\in\underline{X}_{r,h}^{k}$, all $d \in

Theorems & Definitions (25)

  • Remark 1: Choice of component spaces
  • Remark 2: Validity of \ref{['eq:def.pot']}
  • Theorem 3: Conforming lifting
  • proof
  • Remark 4: Piecewise polynomial lifting
  • Remark 5: The lifting is a right-inverse of the interpolator
  • Remark 6: Homogeneous boundary conditions
  • Proposition 7: Primal consistency
  • proof
  • Theorem 8: Adjoint consistency
  • ...and 15 more