Fast Direct Solvers for Integral Equations at Low-Frequency Based on Operator Filtering

TL;DR

This paper introduces a fast direct solver for low-frequency integral equations using operator filtering and low-rank approximations, improving computational efficiency and accuracy in boundary element methods.

Contribution

It presents a novel approach combining operator filtering with low-rank representations to efficiently solve integral equations at low frequencies.

Findings

Effective in reducing spectral errors from boundary element discretizations.

Enables low-rank approximations for compact operator perturbations.

Numerical results demonstrate improved solver performance.

Abstract

This paper focuses on fast direct solvers for integral equations in the low-to-moderate-frequency regime obtained by leveraging preconditioned first kind or second kind operators regularized with Laplacian filters. The spectral errors arising from boundary element discretizations are properly handled by filtering that, in addition, allows for the use of low-rank representations for the compact perturbations of all operators involved. Numerical results show the effectiveness of the approaches and their effectiveness in the direct solution of integral equations.