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.

翻译：我们针对拉普拉斯方程的内外部Dirichlet问题提出了新的第二类积分方程公式。在$\mathbb{R}^d$（$d\geq 2$）中一般Lipschitz域上的$L^2(\Gamma)$空间内（其中$\Gamma$表示域边界），这些公式中的算子既连续又强制。连续性与强制性的这些性质直接意味着：（i）Galerkin方法应用于这些公式时收敛；（ii）随着离散化程度的细化，Galerkin矩阵是良态的，无需算子预条件处理（我们同时证明了关于GMRES收敛性的相应结果）。这些结果的主要意义在于：最近已有证明（参见Chandler-Wilde与Spence, Numer. Math., 150(2):299-271, 2022）存在某些二维与三维Lipschitz域以及三维星形Lipschitz多面体，使得拉普拉斯方程标准第二类积分方程公式（涉及双层势及其伴随算子）中的算子$\textit{无法}$在$L^2(\Gamma)$空间中表示为强制算子与紧算子之和。因此，存在某些二维与三维Lipschitz域以及三维星形Lipschitz多面体，使得$L^2(\Gamma)$空间中的Galerkin方法应用于标准第二类公式时$\textit{不}$收敛，但应用于新公式时$\textit{确实}$收敛。