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Convergence of Calder\'on residuals

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Ralf Hiptmair, Carolina Urzúa-Torres, Anouk Wisse

Numerical Analysis · Submitted 31 Mar 2025

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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.

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arXiv ID2503.23900 (version 1)
Pith IDZVZGSL6M, Pith's permanent record of this paper and its review
CitationAnouk Wisse, Carolina Urz\'ua-Torres, Ralf Hiptmair. "Convergence of Calder\'on residuals." arXiv:2503.23900 (2025). https://pith.science/paper/ZVZGSL6M
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In this paper, we describe a framework to compute expected convergence rates for residuals based on the Calder\'on 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 Calder\'on 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 Calder\'on 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.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 In this paper, we describe a framework to compute expected convergence rates for residuals based on the Calder\'on identities for general second order differential operators for which fundamental solutions are known. contribution 0.99 0.84 0.09 asserted 0.98
s1 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 Calder\'on identities vanish. contribution 0.37 0.21 0.09 hedged possibility 1.00
s2 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. method 0.80 0.82 0.07 asserted 1.00
s3 We test the proposed Calder\'on 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. method 0.86 0.96 0.06 asserted 0.99
s4 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. limitation 0.57 0.95 0.68 asserted 0.97

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from simulation or computation 0.88.

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Original abstract

In this paper, we describe a framework to compute expected convergence rates for residuals based on the Calder\'on 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 Calder\'on 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 Calder\'on 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.

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