Lipschitz 域上 Laplace Dirichlet 问题的强制第二类边界积分方程
Coercive second-kind boundary integral equations for the Laplace Dirichlet problem on Lipschitz domains
AI总结：
针对Lipschitz域上Laplace Dirichlet问题，提出连续且强制的第二类边界积分方程新公式，确保Galerkin方法收敛且矩阵良态，克服了标准公式在某些域上不收敛的缺陷。
摘要：
我们提出了 Laplace 方程内部和外部 Dirichlet 问题的新第二类积分方程公式。这些公式中的算子均在 $\mathbb{R}^d$（$d\geq 2$）中的一般 Lipschitz 域上的空间 $L^2(\Gamma)$ 中连续且强制，其中 $\Gamma$ 表示域的边界。连续性和强制性的这些性质立即意味着：（i）Galerkin 方法应用于这些公式时收敛；（ii）随着离散化细化，Galerkin 矩阵条件数良好，无需算子预处理（并且我们证明了关于 GMRES 收敛的相应结果）。这些结果的主要意义在于，最近已证明（参见 Chandler-Wilde 和 Spence，Numer. Math.，150(2):299-271，2022）存在二维和三维 Lipschitz 域以及三维星形 Lipschitz 多面体，使得 Laplace 方程的标准第二类积分方程公式（涉及双层势及其伴随）中的算子 $\textit{不能}$ 写成 $L^2(\Gamma)$ 空间中强制算子与紧算子之和。因此，存在二维和三维 Lipschitz 域以及三维星形 Lipschitz 多面体，使得 $L^2(\Gamma)$ 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。
英文摘要：
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.
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