Product algebras for Galerkin discretisations of boundary integral operators and their applications

TL;DR

This paper introduces a comprehensive operator algebra for Galerkin discretizations in boundary element methods, simplifying the formulation and solution of boundary integral equations in software applications.

Contribution

It develops a complete operator algebra framework for Galerkin discretizations that accounts for domain, range, and test spaces, enhancing boundary element software capabilities.

Findings

Implementation of operator algebra in Bempp software

Simplified formulation of boundary integral equations

Effective handling of Laplace and Helmholtz problems

Abstract

Operator products occur naturally in a range of regularized boundary integral equation formulations. However, while a Galerkin discretisation only depends on the domain space and the test (or dual) space of the operator, products require a notion of the range. In the boundary element software package Bempp we have implemented a complete operator algebra that depends on knowledge of the domain, range and test space. The aim was to develop a way of working with Galerkin operators in boundary element software that is as close to working with the strong form on paper as possible while hiding the complexities of Galerkin discretisations. In this paper, we demonstrate the implementation of this operator algebra and show, using various Laplace and Helmholtz example problems, how it significantly simplifies the definition and solution of a wide range of typical boundary integral equation problems.