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

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.