Coercive second-kind boundary integral equations for the Laplace Dirichlet problem on Lipschitz domains
Coercive second-kind boundary integral equations for the Laplace Dirichlet problem on Lipschitz domains
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DOI：
10.48550/arxiv.2210.02432
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发表时间：
2022-10
期刊：
影响因子：
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通讯作者：
S. Chandler-Wilde;E. Spence
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文献类型：
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作者：
S. Chandler-Wilde;E. Spence
我们给出了拉普拉斯方程的内外Dirichlet问题的新的第二类积分方程列式。这些公式中的算子在空间$$L^2(\Gamma)$$L 2(Γ)中的$Mathbb{R}^d$$Rd，$$d2$$dΓ中的一般Lipschitz域上是连续的和强制的，其中$$\Gamma$$是域的边界.这些连续性和矫顽力的性质立即意味着：(1)Galerkin方法在应用于这些公式时收敛；(2)Galerkin矩阵在离散化被细化时是良好条件的，不需要算子预条件(我们证明了关于GMRES收敛的相应结果)。这些结果的主要意义在于，最近证明了(参见Chandler-Wilde和Spence在Numer Math150(2)：299-371,2022年)，存在二维和三维的Lipschitz域和三维星形Lipschitz多面体，对于它们，拉普拉斯方程(涉及双层势及其伴随)的标准第二类积分方程式中的算子不能写成$L^2(\Gamma)$L 2(Γ)空间中的强制算子和紧算子的和。因此，存在二维和三维Lipschitz域和三维星形Lipschitz多面体，其中${L^2(\Gamma)}$$L 2(Γ)中的Galerkin方法在应用于标准第二类公式时不收敛，但对于新公式确实收敛。
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 $$\mathbb {R}^d$$ R d , $$d\ge 2$$ d ≥ 2 , in the space $$L^2(\Gamma )$$ L 2 ( Γ ) , where $$\Gamma $$ Γ denotes the boundary of the domain. These properties of continuity and coercivity immediately imply that (1) the Galerkin method converges when applied to these formulations; and (2) 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 in Numer Math 150(2):299–371, 2022) that there exist 2- and 3-d Lipschitz domains and 3-d star-shaped 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) cannot be written as the sum of a coercive operator and a compact operator in the space $$L^2(\Gamma )$$ L 2 ( Γ ) . Therefore there exist 2- and 3-d Lipschitz domains and 3-d star-shaped Lipschitz polyhedra for which Galerkin methods in $${L^2(\Gamma )}$$ L 2 ( Γ ) do not converge when applied to the standard second-kind formulations, but do converge for the new formulations.