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An exterior calculus framework for polytopal methods

Francesco Bonaldi, Daniele A. Di Pietro, Jerome Droniou, Kaibo Hu

TL;DR

The paper develops two polytopal, fully discrete de Rham-type complexes (DDR and VEM) that operate on general polytopal meshes without requiring globally conforming differential-form spaces. It constructs discrete exterior derivatives and potentials via Stokes-type reconstructions, ensuring polynomial consistency and cohomology isomorphism with the continuous de Rham complex. The DDR and VEM frameworks yield stable, convergent schemes for Hodge Laplacian problems and other PDEs while enabling dimensional reductions and flexible meshing. By linking discrete cohomology to the continuous theory and clarifying relationships to FEEC/FES and DDF, the work provides a robust, scalable route for high-order, topology-aware discretizations on polytopal grids.

Abstract

We develop in this work the first polytopal complexes of differential forms. These complexes, inspired by the Discrete De Rham and the Virtual Element approaches, are discrete versions of the de Rham complex of differential forms built on meshes made of general polytopal elements. Both constructions benefit from the high-level approach of polytopal methods, which leads, on certain meshes, to leaner constructions than the finite element method. We establish commutation properties between the interpolators and the discrete and continuous exterior derivatives, prove key polynomial consistency results for the complexes, and show that their cohomologies are isomorphic to the cohomology of the continuous de Rham complex.

An exterior calculus framework for polytopal methods

TL;DR

The paper develops two polytopal, fully discrete de Rham-type complexes (DDR and VEM) that operate on general polytopal meshes without requiring globally conforming differential-form spaces. It constructs discrete exterior derivatives and potentials via Stokes-type reconstructions, ensuring polynomial consistency and cohomology isomorphism with the continuous de Rham complex. The DDR and VEM frameworks yield stable, convergent schemes for Hodge Laplacian problems and other PDEs while enabling dimensional reductions and flexible meshing. By linking discrete cohomology to the continuous theory and clarifying relationships to FEEC/FES and DDF, the work provides a robust, scalable route for high-order, topology-aware discretizations on polytopal grids.

Abstract

We develop in this work the first polytopal complexes of differential forms. These complexes, inspired by the Discrete De Rham and the Virtual Element approaches, are discrete versions of the de Rham complex of differential forms built on meshes made of general polytopal elements. Both constructions benefit from the high-level approach of polytopal methods, which leads, on certain meshes, to leaner constructions than the finite element method. We establish commutation properties between the interpolators and the discrete and continuous exterior derivatives, prove key polynomial consistency results for the complexes, and show that their cohomologies are isomorphic to the cohomology of the continuous de Rham complex.
Paper Structure (49 sections, 17 theorems, 178 equations, 2 tables)

This paper contains 49 sections, 17 theorems, 178 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Setting
  3. Spaces of differential forms
  4. Integration by parts
  5. Hodge star
  6. Polytopal mesh
  7. Local polynomial spaces of differential forms
  8. Trimmed local polynomial spaces
  9. Discrete de Rham complex
  10. Discrete spaces
  11. Interpolators and interpretation of the polynomial components
  12. Local discrete potentials and discrete exterior derivative
  13. Global discrete exterior derivative and DDR complex
  14. Discrete $L^2$-products
  15. Application to the Hodge Laplacian
  16. ...and 34 more sections

Key Result

Lemma 1

Let $(k,d)$ be integers such that $k\le d\le n$, $f\in\Delta_d(\mathcal{M}_h)$, and $\mathcal{X}$ be a closed subspace of $L^2\Lambda^{d-k}(f)$. Then, it holds: For all $\omega\in L^2\Lambda^k(f)$ and all $\mu\in\mathcal{X}$,

Theorems & Definitions (60)

  • Lemma 1: Projectors on subspaces of differential forms
  • proof
  • Example 2: Interpretation in terms of vector proxies
  • Example 3: Interpretation of \ref{['eq:trimmed.spaces:ell>=1']} in terms of vector proxies
  • Lemma 4: Traces of trimmed polynomial spaces
  • proof
  • Remark 5: Choice of polynomial components
  • Remark 6: Virtual spaces
  • Remark 7: Comparison with trimmed finite element sequences
  • Remark 8: Boundary integration and orientation
  • ...and 50 more