论文标题
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
论文作者
论文摘要
我们为Laplace方程的内部和外部Dirichlet问题提供了新的第二种积分方程式。这些配方中的操作员在$ \ mathbb {r}^d $，$ d \ geq 2 $中的一般Lipschitz域上既连续又是强制性的，在太空$ l^2（γ）$中，$γ$表示域的边界。这些连续性和强制性的特性立即表明（i）galerkin方法在应用于这些配方时会收敛； （ii）由于离散化的精制而无需操作员预处理，因此对Galerkin矩阵进行了充分的条件（我们证明了有关GMRE的收敛性的相应结果）。 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 {不能} $写为“强制操作员的总和”和“ space $ l^2（γ）$”中的紧凑型操作员。因此，存在2个和3-D Lipschitz域和3-D星状Lipschitz Polyhedra，用于$ l^2（γ）$ do $ \ textit {not} $中的Galerkin方法当应用于标准的第二键配方时，但是$ \ textit {do} $ commented for New New New Fiverlection。
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.
