Existence of $\mathcal{H}$-matrix approximants to the inverse of BEM matrices: the hyper-singular integral operator

TL;DR

This paper demonstrates that inverses of matrices from discretized hyper-singular integral operators on closed surfaces can be efficiently approximated by blockwise low-rank matrices, including $ ext{H}$-matrices and hierarchical Cholesky factors, at an exponential rate.

Contribution

It proves the exponential approximability of inverse matrices of hyper-singular operators in $ ext{H}$-matrix format for two discretization methods, including symmetric positive definite cases.

Findings

Inverse matrices can be approximated exponentially fast in block rank.

Hierarchical Cholesky factors are also approximable at an exponential rate.

Results apply to both saddle point and stabilized discretizations.

Abstract

We consider discretizations of the hyper-singular integral operator on closed surfaces and show that the inverses of the corresponding system matrices can be approximated by blockwise low-rank matrices at an exponential rate in the block rank. We cover in particular the data-sparse format of $\mathcal{H}$-matrices. We show the approximability result for two types of discretizations. The first one is a saddle point formulation, which incorporates the constraint of vanishing mean of the solution. The second discretization is based on a stabilized hyper-singular operator, which leads to symmetric positive definite matrices. In this latter setting, we also show that the hierarchical Cholesky factorization can be approximated at an exponential rate in the block rank.