A fast direct solver for boundary value problems on locally perturbed geometries

Abstract

Many applications involve solving several boundary value problems on geometries that are local perturbations of an original geometry. The boundary integral equation for a problem on a locally perturbed geometry can be expressed as a low rank update to the original system. A fast direct solver for the new linear system is presented in this paper. The solution technique utilizes a precomputed fast direct solver for the original geometry to efficiently create the low rank factorization of the update matrix and to accelerate the application of the Sherman-Morrison formula. The method is ideally suited for problems where the local perturbation is the same but its placement on the boundary changes and problems where the local perturbation is a refined discretization on the same geometry. Numerical results illustrate that for fixed local perturbation the method is three times faster than building a new fast direct solver from scratch.