Product algebras for Galerkin discretisations of boundary integral operators and their applications
This work provides a practical tool for researchers and engineers using boundary element methods, making it easier to work with operator products in Galerkin discretizations, though it is an incremental improvement in software implementation.
The paper presents an operator algebra for Galerkin discretizations of boundary integral operators, implemented in the Bempp software, which simplifies the definition and solution of boundary integral equation problems by handling domain, range, and test spaces. The approach is demonstrated on Laplace and Helmholtz problems, showing significant simplification in problem setup.
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.