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Broken-FEEC discretizations and Hodge Laplace problems

Martin Campos-Pinto, Yaman Güçlü

TL;DR

The paper advances CONGA-style broken FEEC discretizations for Hilbert complexes with possible nontrivial harmonic fields by constructing dual commuting projections and a stabilized CONGA Hodge Laplacian that maintains the same discrete harmonic kernel as the conforming counterpart. It provides a rigorous a priori error framework and spectral convergence results under assumptions on conforming-projection stability and moment preservation, and demonstrates the locality of both primal and dual operators for polynomial tensor-product elements. The analysis covers source and eigenvalue problems and highlights the role of stabilization in achieving numerical robustness, including spectral correctness and avoidance of spurious modes. Numerical experiments on a square domain corroborate the theory, showing comparable convergence to conforming schemes and illustrating penalization effects on eigenvalues; the work also points to extensions to spline-based, multi-patch domains and Hamiltonian particle discretizations.

Abstract

This article studies structure-preserving discretizations of Hilbert complexes with nonconforming spaces that rely on projections onto an underlying conforming subcomplex. This approach follows the conforming/nonconforming Galerkin (CONGA) method introduced in [doi.org/10.1090/mcom/3079, doi.org/10.5802/smai-jcm.20, doi.org/10.5802/smai-jcm.21] to derive efficient structure-preserving finite element schemes for the time-dependent Maxwell and Maxwell-Vlasov systems by relaxing the curl-conforming constraint in finite element exterior calculus (FEEC) spaces. Here, it is extended to the discretization of full Hilbert complexes with possibly nontrivial harmonic fields, and the properties of the CONGA Hodge Laplacian operator are investigated. By using block-diagonal mass matrices which may be locally inverted, this framework possesses a canonical sequence of dual commuting projection operators which are local, and it naturally yields local discrete coderivative operators, in contrast to conforming FEEC discretizations. The resulting CONGA Hodge Laplacian operator is also local, and its kernel consists of the same discrete harmonic fields as the underlying conforming operator, provided that a symmetric stabilization term is added to handle the space nonconformities. Under the assumption that the underlying conforming subcomplex admits a bounded cochain projection, and that the conforming projections are stable with moment-preserving properties, a priori convergence results are established for both the CONGA Hodge Laplace source and eigenvalue problems. Our theory is finally illustrated with a spectral element method, and numerical experiments are performed which corroborate our results. Applications to spline finite elements on multi-patch mapped domains are described in a related article [arXiv:2208.05238] for which the present work provides a theoretical background.

Broken-FEEC discretizations and Hodge Laplace problems

TL;DR

The paper advances CONGA-style broken FEEC discretizations for Hilbert complexes with possible nontrivial harmonic fields by constructing dual commuting projections and a stabilized CONGA Hodge Laplacian that maintains the same discrete harmonic kernel as the conforming counterpart. It provides a rigorous a priori error framework and spectral convergence results under assumptions on conforming-projection stability and moment preservation, and demonstrates the locality of both primal and dual operators for polynomial tensor-product elements. The analysis covers source and eigenvalue problems and highlights the role of stabilization in achieving numerical robustness, including spectral correctness and avoidance of spurious modes. Numerical experiments on a square domain corroborate the theory, showing comparable convergence to conforming schemes and illustrating penalization effects on eigenvalues; the work also points to extensions to spline-based, multi-patch domains and Hamiltonian particle discretizations.

Abstract

This article studies structure-preserving discretizations of Hilbert complexes with nonconforming spaces that rely on projections onto an underlying conforming subcomplex. This approach follows the conforming/nonconforming Galerkin (CONGA) method introduced in [doi.org/10.1090/mcom/3079, doi.org/10.5802/smai-jcm.20, doi.org/10.5802/smai-jcm.21] to derive efficient structure-preserving finite element schemes for the time-dependent Maxwell and Maxwell-Vlasov systems by relaxing the curl-conforming constraint in finite element exterior calculus (FEEC) spaces. Here, it is extended to the discretization of full Hilbert complexes with possibly nontrivial harmonic fields, and the properties of the CONGA Hodge Laplacian operator are investigated. By using block-diagonal mass matrices which may be locally inverted, this framework possesses a canonical sequence of dual commuting projection operators which are local, and it naturally yields local discrete coderivative operators, in contrast to conforming FEEC discretizations. The resulting CONGA Hodge Laplacian operator is also local, and its kernel consists of the same discrete harmonic fields as the underlying conforming operator, provided that a symmetric stabilization term is added to handle the space nonconformities. Under the assumption that the underlying conforming subcomplex admits a bounded cochain projection, and that the conforming projections are stable with moment-preserving properties, a priori convergence results are established for both the CONGA Hodge Laplace source and eigenvalue problems. Our theory is finally illustrated with a spectral element method, and numerical experiments are performed which corroborate our results. Applications to spline finite elements on multi-patch mapped domains are described in a related article [arXiv:2208.05238] for which the present work provides a theoretical background.

Paper Structure

This paper contains 22 sections, 14 theorems, 189 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Hilbert complexes and FEEC discretizations
  3. Hodge Laplacian operator
  4. Conforming FEEC discretization
  5. Discrete Hodge Laplacian operator
  6. Broken FEEC discretization
  7. Projection-based differential operators
  8. Commuting diagrams for strong and weak broken FEEC complexes
  9. Broken Hodge-Helmholtz decompositions
  10. Analysis of broken-FEEC Hodge Laplace problems
  11. The Hodge Laplace source problem
  12. A priori error analysis
  13. The Hodge Laplace eigenvalue problem
  14. Application to polynomial finite elements
  15. Tensor-product local spaces
  16. ...and 7 more sections

Key Result

Theorem 2.2

If the conforming discrete complex $(V^c_h, d)$ admits a $V$-bounded cochain projection pih, then the discrete Poincaré inequality pcr_hc holds with a constant $c_{P,h} = c_{P} \lVert\pi_h\rVert_V$ where $c_{P}$ is from pcr. Moreover, for any $(\sigma, u, p) \in X^c_h = V^{\ell-1,c}_h \times V^{\ell holds for some $\gamma > 0$ depending only on the discrete Poincaré constant $c_{P,h}$. In particul

Figures (4)

  • Figure 1: Convergence curves for the Hodge Laplace source problem, discretized with a conforming (left) and broken FEEC (right) method. For the latter, various penalization regimes lead to similar curves, as described in the text. On the horizontal axis we report the number of cells $K$ along each direction, hence $h = 2\pi/K$; on the vertical axis we report the $L^2$ norm of the error, divided by the $L^2$ norm of the exact solution.
  • Figure 2: Discrete eigenvalues of the CONGA Hodge Laplacian operator on the square, with strong (left) and weak (right) penalization regimes, as discussed in the text.
  • Figure 3: Eigenvalue errors for the CONGA Hodge Laplacian operator on the square, for the same parameters as in Figure \ref{['fig:HL_eigs']}.
  • Figure 4: Positive eigenvalues and errors for the unpenalized ($\alpha_h = 0$) CONGA Hodge Laplacian operator on the square.

Theorems & Definitions (35)

  • Remark 2.1: discretization parameter $h$
  • Theorem 2.2
  • Theorem 3.2
  • Theorem 3.3
  • Remark 3.4
  • proof : Proof.
  • Corollary 3.5
  • proof : Proof of Theorem \ref{['th:HL_ker']}.
  • Lemma 3.6
  • proof : Proof.
  • ...and 25 more