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Broken-FEEC on multipatch domains with local refinements

Martin Campos Pinto, Frederik Schnack

TL;DR

The paper tackles the challenge of performing structure-preserving simulations on locally refined multipatch domains with nonmatching interfaces. It introduces explicit, moment-preserving conforming projection operators that enforce $H^1$ and $H(curl)$ continuity across interfaces while preserving high-order polynomial moments, enabling broken-FEEC de Rham sequences to retain accurate derivatives. This leads to energy-preserving, explicit Maxwell solvers with virtually no spurious modes, as demonstrated by numerical experiments across Poisson, Maxwell, curl-curl eigenvalue, and Helmholtz problems. The results show substantial reductions in spurious waves and efficient handling of local refinements, highlighting the practical impact for electromagnetic and related PDE simulations on complex geometries.

Abstract

This article introduces a novel approach for broken-FEEC (Finite Element Exterior Calculus), extending its application to locally refined spline spaces with non-matching interfaces. Traditional broken-FEEC allows for discontinuous discretizations at patch interfaces, preserving the de Rham structure and offering computational benefits. However, local refinements often lead to numerical artifacts. Our solution involves developing moment-preserving discrete conforming projection operators. These operators are explicit, localized, and metric-independent, ensuring $H^1$ and $H(\text{curl})$ continuity across non-matching interfaces while preserving high-order polynomial moments. This results in broken-FEEC de Rham sequences with accurate strong and weak derivatives, leading to energy-preserving Maxwell solvers that are explicit and virtually free of spurious modes. Numerical simulations confirm the efficacy of our method in eliminating spurious waves.

Broken-FEEC on multipatch domains with local refinements

TL;DR

The paper tackles the challenge of performing structure-preserving simulations on locally refined multipatch domains with nonmatching interfaces. It introduces explicit, moment-preserving conforming projection operators that enforce and continuity across interfaces while preserving high-order polynomial moments, enabling broken-FEEC de Rham sequences to retain accurate derivatives. This leads to energy-preserving, explicit Maxwell solvers with virtually no spurious modes, as demonstrated by numerical experiments across Poisson, Maxwell, curl-curl eigenvalue, and Helmholtz problems. The results show substantial reductions in spurious waves and efficient handling of local refinements, highlighting the practical impact for electromagnetic and related PDE simulations on complex geometries.

Abstract

This article introduces a novel approach for broken-FEEC (Finite Element Exterior Calculus), extending its application to locally refined spline spaces with non-matching interfaces. Traditional broken-FEEC allows for discontinuous discretizations at patch interfaces, preserving the de Rham structure and offering computational benefits. However, local refinements often lead to numerical artifacts. Our solution involves developing moment-preserving discrete conforming projection operators. These operators are explicit, localized, and metric-independent, ensuring and continuity across non-matching interfaces while preserving high-order polynomial moments. This results in broken-FEEC de Rham sequences with accurate strong and weak derivatives, leading to energy-preserving Maxwell solvers that are explicit and virtually free of spurious modes. Numerical simulations confirm the efficacy of our method in eliminating spurious waves.

Paper Structure

This paper contains 33 sections, 8 theorems, 159 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Broken-FEEC
  3. Multipatch spline finite element spaces
  4. Tensor-product and mapped spline bases
  5. Conforming constraints on patch interfaces
  6. Weak differential operators in conforming FEEC sequences
  7. Weak broken differential operators and moment preservation
  8. Broken-FEEC discretizations of model PDEs
  9. Broken degrees of freedom
  10. Weak differential operators
  11. Poisson problem
  12. Time-harmonic Maxwell problem
  13. Curl-curl eigenvalue problem
  14. Time-domain Maxwell
  15. Time-domain Helmholtz
  16. ...and 18 more sections

Key Result

Theorem 2.3

Let $\tilde{\Pi}^\ell_h$ be the $L^2$-projection onto the conforming space $V^\ell_h$ without boundary conditions, and let be the discrete differential operators corresponding to div_curl_h0. Then the following diagram commutes: \begin{tikzcd}[row sep=large, column sep=large] V^0_{h} \arrow[shift left]{r}{\bgrad} & V^1_{h} \arrow[shift left]{r}{\curl} \arrow[shift left]{l}{\Div_{h, 0}} & \arro

Figures (13)

  • Figure 1: The edge-local logical domain $\hat{\Omega}(\mathrm{e})$ (right) and the edge-local physical domain $\Omega(\mathrm{e})$ (left) for a horizontal edge $\mathrm{e}$, different resolutions and the mapping $F(\mathrm{e})$. Additionally, we show the orientation of the logical variables and indicate the direction of increasing basis index ordering.
  • Figure 2: Different refinements of the curved L-shaped domain.
  • Figure 3: Convergence curves for the weak divergence test-case with moment preservation of order $p+1$ (left) and without (right). We plot the relative $L^2$-error of the weak divergence operator for different polynomial degrees and refinement levels on the curved L-shaped domains in Figure \ref{['fig:weak_diff_grids']}.
  • Figure 4: Convergence curves for the weak $\mathop{\mathrm{\mathbf{curl}}}\nolimits$ test-case with moment preservation of order $p+1$ (left) and without (right). We plot the relative $L^2$-error of the weak $\mathop{\mathrm{\mathbf{curl}}}\nolimits$ operator for different polynomial degrees and refinement levels on the curved L-shaped domains in Figure \ref{['fig:weak_diff_grids']}.
  • Figure 5: The non-matching refinements of the top-right handle of the pretzel domain.
  • ...and 8 more figures

Theorems & Definitions (23)

  • Theorem 2.3
  • Theorem 2.4
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7
  • Proposition 3.1
  • Proposition 3.2
  • Theorem 3.3
  • proof
  • Remark 3.4
  • ...and 13 more