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Convergence of Calderón residuals

Ralf Hiptmair, Carolina Urzúa-Torres, Anouk Wisse

TL;DR

The authors introduce a framework to predict convergence rates for Calderón residuals derived from Calderón identities for general second-order PDEs with known fundamental solutions, enabling practical code validation for boundary integral operators using simple vector-norm residuals and manufactured solutions. They apply the framework to Laplace and time-harmonic Maxwell problems, implement the tests in Bempp, and examine residual behavior under mesh refinement and artificial perturbations of Galerkin matrices and quadrature. The experiments show Calderón residuals can reveal implemented errors in many but not all cases; they often exhibit convergence and can outperform standard MMS tests for certain Maxwell scenarios, but their lack of sharpness in some Laplace demonstrations motivates future sharpening. Overall, the work provides a tangible debugging tool for BEM practitioners while outlining limitations and directions for improving the sharpness and robustness of Calderón-residual-based validation.

Abstract

In this paper, we describe a framework to compute expected convergence rates for residuals based on the Calderón identities for general second order differential operators for which fundamental solutions are known. The idea is that these rates could be used to validate implementations of boundary integral operators and allow to test operators separately by choosing solutions where parts of the Calderón identities vanish. Our estimates rely on simple vector norms, and thus avoid the use of hard-to-compute norms and the residual computation can be easily implemented in existing boundary element codes. We test the proposed Calderón residuals as debugging tool by introducing artificial errors into the Galerkin matrices of some of the boundary integral operators for the Laplacian and time-harmonic Maxwell's equations. From this, we learn that our estimates are not sharp enough to always detect errors, but still provide a simple and useful debugging tool in many situations.

Convergence of Calderón residuals

TL;DR

The authors introduce a framework to predict convergence rates for Calderón residuals derived from Calderón identities for general second-order PDEs with known fundamental solutions, enabling practical code validation for boundary integral operators using simple vector-norm residuals and manufactured solutions. They apply the framework to Laplace and time-harmonic Maxwell problems, implement the tests in Bempp, and examine residual behavior under mesh refinement and artificial perturbations of Galerkin matrices and quadrature. The experiments show Calderón residuals can reveal implemented errors in many but not all cases; they often exhibit convergence and can outperform standard MMS tests for certain Maxwell scenarios, but their lack of sharpness in some Laplace demonstrations motivates future sharpening. Overall, the work provides a tangible debugging tool for BEM practitioners while outlining limitations and directions for improving the sharpness and robustness of Calderón-residual-based validation.

Abstract

In this paper, we describe a framework to compute expected convergence rates for residuals based on the Calderón identities for general second order differential operators for which fundamental solutions are known. The idea is that these rates could be used to validate implementations of boundary integral operators and allow to test operators separately by choosing solutions where parts of the Calderón identities vanish. Our estimates rely on simple vector norms, and thus avoid the use of hard-to-compute norms and the residual computation can be easily implemented in existing boundary element codes. We test the proposed Calderón residuals as debugging tool by introducing artificial errors into the Galerkin matrices of some of the boundary integral operators for the Laplacian and time-harmonic Maxwell's equations. From this, we learn that our estimates are not sharp enough to always detect errors, but still provide a simple and useful debugging tool in many situations.

Paper Structure

This paper contains 21 sections, 5 theorems, 51 equations, 2 figures, 26 tables.

Table of Contents

  1. Introduction
  2. General Case
  3. First Application: Laplacian
  4. Implementation and sanity checks
  5. Empiric convergence of norms of basis functions
  6. Checking convergence of Calderón residuals
  7. Numerical results when introducing artificial errors
  8. Inaccurate quadrature
  9. Perturbing entries in $\mathbf{V}_h$
  10. Perturbing entries in $\mathbf{W}_h$
  11. Second Application: Time-Harmonic Maxwell's Equations
  12. Implementation and sanity checks
  13. Empiric convergence of norms of basis functions
  14. Checking convergence of Calderón residuals
  15. Numerical results when introducing artificial errors
  16. ...and 6 more sections

Key Result

Theorem 3.3

Let $\Gamma$ fulfill either of the conditions in Assumption assumption_Gamma. Then, we have for the interpolation $\mathcal{I}_h^{p-1,-1}: L^2(\Gamma) \to \mathcal{ S}_{\mathcal{G}}^{p-1,-1}$ and $0\leq t\leq s \leq p$ and all $u\in H^s_{\text{pw}}(\Gamma)$ (where this space is as defined in Sauter_

Figures (2)

  • Figure 1: Eigenvalues when scaling the diagonal of $\mathbf{V}_h$ by $1/2$ in Example 1a for the Laplacian (Error B). Results for smallest mesh, i.e., $N= 128$ elements.
  • Figure 2: Eigenvalues when scaling the diagonal of $\mathbf{V}_h$ by $1/2$ in Example 2 for the Laplacian (Error B). Results for smallest mesh, i.e., $N= 84$ elements.

Theorems & Definitions (11)

  • Remark
  • Definition 3.1: Sauter_Schwab
  • Theorem 3.3: Sauter_Schwab
  • Theorem 3.4: Sauter_Schwab
  • Theorem 3.5
  • proof
  • Lemma 4.1
  • Theorem 4.2
  • proof
  • Remark
  • ...and 1 more