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

## TL;DR

This paper introduces a fast direct solver for boundary value problems on geometries with local perturbations, leveraging low rank updates and precomputed solutions to significantly reduce computation time.

## Contribution

It presents a novel method that efficiently updates solutions for perturbed geometries using low rank updates and the Sherman-Morrison formula, saving computational effort.

## Key findings

- Solver is three times faster than building from scratch for fixed perturbations.
- Method effectively handles local boundary modifications and refined discretizations.
- Numerical results demonstrate significant computational savings.

## 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.