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

TL;DR

The paper develops coercive, second-kind boundary-integral formulations for the Laplace interior and exterior Dirichlet problems on general Lipschitz domains, proving continuity and coercivity of the new operators in $L^2(\Gamma)$ and guaranteeing Galerkin convergence for any asymptotically-dense subspace. It further shows that, after diagonal preconditioning, the GMRES iterations remain uniformly bounded as the discretisation refines, removing the need for operator preconditioning in many settings. The authors also extend the framework to 2D Lipschitz domains and oblique Robin problems, and provide Helmholtz exterior Dirichlet problem formulations with analogous coercivity properties, situating the work within the broader boundary-integral-method literature and addressing previously known limitations. Overall, the results deliver robust, coercive $L^2(\Gamma)$ boundary-integral formulations that enable stable, high-order Galerkin/BEM implementations on rough geometries, including star-shaped and Lipschitz polyhedral domains, with practical implications for large-scale simulations.

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 $L^2(Γ)$, where $Γ$ 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 $L^2(Γ)$. Therefore there exist 2- and 3-d Lipschitz domains and 3-d starshaped Lipschitz polyhedra for which Galerkin methods in $L^2(Γ)$ do $\textit{not}$ converge when applied to the standard second-kind formulations, but $\textit{do}$ converge for the new formulations.

Figures (1)

• Figure 1.1: Views from above and below of the open-book polyhedron $\Omega_{\theta,n}$ of ChSp:22, with $n=4$ pages and opening angle $\theta = \pi/2$

Theorems & Definitions (77)

• Theorem 1.1: The main abstract theorem on convergence of the Galerkin method.
• Theorem 1.2
• Definition 1.3: Laplace interior Dirichlet problem (IDP)
• Definition 1.4: Laplace exterior Dirichlet problem (EDP)
• Definition 1.5: Laplace IDP formulated via non-tangential limits
• Definition 1.6: Laplace EDP formulated via non-tangential limits
• Theorem 1.7: Equivalence of the different formulations of the IDP and EDP
• Theorem 2.1: New integral equations for Laplace IDP with $d\geq 3$
• Theorem 2.2: New integral equations for Laplace EDP with $d\geq 3$
• Definition 2.3