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

TL;DR

This paper introduces new second-kind boundary integral equations for Laplace's Dirichlet problem on Lipschitz domains that are both continuous and coercive, ensuring convergence and well-conditioning of numerical methods where standard formulations fail.

Contribution

The authors develop novel integral-equation formulations that are coercive on Lipschitz domains, guaranteeing convergence and good conditioning of Galerkin methods without operator preconditioning.

Findings

New formulations are continuous and coercive on Lipschitz domains.

Galerkins method converges with these formulations, unlike standard ones.

Matrices are well-conditioned as discretization refines.

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 $\mathbb{R}^d$, $d\geq 2$, 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 starshaped 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) $\textit{cannot}$ 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 starshaped Lipschitz polyhedra for which Galerkin methods in $L^2(\Gamma)$ do $\textit{not}$ converge when applied to the standard second-kind formulations, but $\textit{do}$ converge for the new formulations.