On the space-time discretization of variational retarded potential boundary integral equations

TL;DR

This paper develops and analyzes space-time boundary element methods for solving the wave equation in three dimensions, focusing on retarded potentials, quadrature techniques, and efficient implementation, validated through numerical experiments.

Contribution

It introduces a novel space-time discretization approach for retarded potential boundary integral equations with new quadrature techniques and implementation strategies.

Findings

Effective quadrature methods for retarded layer potentials.

Numerical experiments demonstrate the methods' capacity.

Insights into efficient implementation of retarded potentials.

Abstract

This paper discusses the practical development of space-time boundary element methods for the wave equation in three spatial dimensions. The employed trial spaces stem from simplex meshes of the lateral boundary of the space-time cylinder. This approach conforms genuinely to the distinguished structure of the solution operators of the wave equation, so-called retarded potentials. Since the numerical evaluation of the arising integrals is intricate, the bulk of this work is constituted by ideas about quadrature techniques for retarded layer potentials and associated energetic bilinear forms. Finally, we glimpse at algorithmic aspects regarding the efficient implementation of retarded potentials in the space-time setting. The proposed methods are verified by means of numerical experiments, which illustrate their capacity.