An iterative substructuring method for the \(p\)-version of the boundary element method for hypersingular integral operators in three dimensions
The aim of this paper is to investigate preconditioners for the stiffness matrix of a linear system, which provide good bounds for the spectral condition number \(k\). The study of the preconditioners is based on domain decomposition. As a model problem the weak form of the hypersingular integral equation \[ \langle Du,v\rangle _{L^{2}(\Gamma)}=\langle f,v\rangle _{L^{2}(\Gamma)} \] is considered on the plane rectangular screen \(\Gamma \subset \mathbb{R}^{3}\) where \(f\in \widetilde{H}^{-1/2}(\Gamma)\) is a given function. Here \(D\) is the hypersingular integral operator \[ Du(x)=\frac{1}{4\pi } \frac{\partial }{\partial n_{x}}\oint\limits_{\Gamma }u(y) \frac{\partial }{\partial n_{y}} \frac{1}{| x-y| }dS_{y},\;x\in \Gamma \] which is a continuous and positive definite mapping from \(\widetilde{H} ^{1/2}(\Gamma)\) onto \(\widetilde{H}^{-1/2}(\Gamma).\) In the last section some numerical experiments are presented with three different preconditioners defined during the paper for the \(p\)-version of the boundary element method.
scientific article; zbMATH DE number 1187547