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

Chandler-Wilde, S. N. ORCID: https://orcid.org/0000-0003-0578-1283 and Spence, E. A. (2024) Coercive second-kind boundary integral equations for the Laplace Dirichlet problem on Lipschitz domains. Numerische Mathematik, 156 (4). pp. 1325-1384. ISSN 0029-599X doi: 10.1007/s00211-024-01424-9

Abstract/Summary

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 R^d, d>=2, in the space L^2(\Gamma), where \Gamma denotes the boundary of the domain. These properties of continuity and coercivity immediately imply that (i) the Galerkin method converges when applied to these formulations; and (ii) 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, Numer. Math., 150(2):299-271, 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 L^2(\Gamma). Therefore there exist 2- and 3-d Lipschitz domains and 3-d star-shaped Lipschitz polyhedra for which Galerkin methods in L^2(\Gamma) do not converge when applied to the standard second-kind formulations, but do converge for the new formulations.

Item Type	Article
URI	https://reading-clone.eprints-hosting.org/id/eprint/116628
Identification Number/DOI	10.1007/s00211-024-01424-9
Refereed	Yes
Divisions	Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
Publisher	Springer
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