BEM for variable coefficient second-order problems

TL;DR

This paper introduces a boundary element method that handles variable coefficient second-order elliptic problems without relying on explicit fundamental solutions, enabling efficient discretisation and matrix compression.

Contribution

It develops a novel BEM approach that constructs boundary operators from Galerkin discretisations, extending applicability to strongly elliptic operators with variable coefficients.

Findings

Removes dependence on explicit fundamental solutions.

Retains dimension reduction and compatibility with data-sparse matrices.

Enables efficient approximation of boundary operators from finite element discretisations.

Abstract

A novel boundary element method (BEM) removes the classical dependence on explicit fundamental solutions and extends quasi-optimal BEM discretisations to strongly elliptic operators with variable coefficients. The approach constructs a computable approximation of the boundary operator from a Galerkin discretisation of the underlying elliptic differential operator in a one-time preprocessing step, for instance by conforming finite elements. The resulting algebraic formulation retains the dimension reduction intrinsic to boundary integral methods and is compatible with standard data-sparse matrix compression techniques.