Coercive second-kind boundary integral equations for the Laplace Dirichlet problem on Lipschitz domains

Simon Chandler-Wilde, Euan Spence

Research output: Contribution to journalArticlepeer-review

Abstract

We present new second-kind integral-equation formulations of the interior and exterior Dirichlet problems for Laplace’s equation. The operators in these formulations are both continuous and coercive on general Lipschitz domains in , , in the space , where denotes the boundary of the domain. These properties of continuity and coercivity immediately imply that (1) the Galerkin method converges when applied to these formulations; and (2) the Galerkin matrices are well-conditioned as the discretisation is refined, without the need for operator preconditioning (and we prove a corresponding result about the convergence of GMRES). The main significance of these results is that it was recently proved (see Chandler-Wilde and Spence in Numer Math 150(2):299–371, 2022) that there exist 2- and 3-d Lipschitz domains and 3-d star-shaped Lipschitz polyhedra for which the operators in the standard second-kind integral-equation formulations for Laplace’s equation (involving the double-layer potential and its adjoint) cannot be written as the sum of a coercive operator and a compact operator in the space . Therefore there exist 2- and 3-d Lipschitz domains and 3-d star-shaped Lipschitz polyhedra for which Galerkin methods in do not converge when applied to the standard second-kind formulations, but do converge for the new formulations.
Original languageEnglish
Number of pages60
JournalNumerische Mathematik
Early online date18 Jun 2024
DOIs
Publication statusPublished - 18 Jun 2024

Data Availability Statement

No new data were created in this study.

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