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Convergence of Calderón residuals

by R. Hiptmair and C. Urzúa-Torres and A. Wisse

(Report number 2025-10)

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.

Keywords: boundary element method, Galerkin method, Calderon identities, a priori convergence estimates

@Techreport{HUW25_1131,
  author = {R. Hiptmair and C. Urzúa-Torres and A. Wisse},
  title = {Convergence of Calderón residuals},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2025-10},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2025/2025-10.pdf },
  year = {2025}
}

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