Product Algebras for Galerkin Discretisations of Boundary Integral Operators and their Applications

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.

Weak Imposition of Signorini Boundary Conditions on the Boundary Element Method

“…This leads us to Algorithm 6.1, an iterative method for solving the contact problem. In all of the computations in this section, we preconditioned the GMRES solver using a mass matrix preconditioner applied blockwise from the left, as described in [3]. Algorithm 6.1.…”

“…For the problems discussed in [2], we have run numerical experiments using this space pairing and observe the full order 3 2 convergence in a low number of iterations. A deeper investigation of this method using these dual spaces, and the adaption of the theory to this case, warrants future work.…”

Calderón preconditioning of PMCHWT boundary integral equations for scattering by multiple absorbing dielectric particles

“…Instead one needs to introduce appropriate mass matrices, which arise in the strong form of the Galerkin operators. For further details about weak and strong discrete forms we refer to [55]; here we just outline the details relevant to the problem at hand.…”

“…In this paper, we investigate the performance of Calderón preconditioning strategies in the context of light scattering by single and multiple complex ice crystals. We also investigate another operator-based preconditioning strategy, "mass-matrix preconditioning", which uses the discrete strong form of the PMCHWT operator [54,55] and does not require a second application of the PM-CHWT operator. For the case of scattering by multiple particles we consider three preconditioning strategies: mass matrix preconditioning; full Calderón preconditioning, involving a second application of the PMCHWT operator; and diagonal Calderón preconditioning, in which only the self-interaction blocks are preconditioned.…”

Bempp-cl: A fast Python based just-in-time compiling boundary element library.

“…Bempp-cl provides a comprehensive collection of routines for the assembly of boundary integral operators to solve a wide range of relevant application problems. It contains an operator algebra that allows a straight-forward implementation of complex operator preconditioned systems of boundary integral equations (Betcke et al, 2020) and in particular implements everything that is required for Calderón preconditioned Maxwell (Scroggs et al, 2017) problems. Bemppcl uses PyOpenCL (Klöckner et al, 2012) to just-in-time compile its computational kernels on a wide range of CPU and GPU devices and modern architectures.…”