Boundary Element Methods for the Wave Equation based on Hierarchical Matrices and Adaptive Cross Approximation

TL;DR

This paper introduces a novel approximation scheme for time-domain Boundary Element Methods applied to wave scattering problems, significantly reducing storage and computational costs while maintaining accuracy.

Contribution

It combines $\\mathcal{H}^2$-matrix compression with adaptive cross approximation in the frequency domain for efficient wave equation solutions.

Findings

Reduced storage requirements for BEM matrices

Significant decrease in computational costs

Preserved accuracy of the wave scattering solutions

Abstract

Time-domain Boundary Element Methods (BEM) have been successfully used in acoustics, optics and elastodynamics to solve transient problems numerically. However, the storage requirements are immense, since the fully populated system matrices have to be computed for a large number of time steps or frequencies. In this article, we propose a new approximation scheme for the Convolution Quadrature Method (CQM) powered BEM, which we apply to scattering problems governed by the wave equation. We use $\mathcal{H}^2$-matrix compression in the spatial domain and employ an adaptive cross approximation (ACA) algorithm in the frequency domain. In this way, the storage and computational costs are reduced significantly, while the accuracy of the method is preserved.