Stable least-squares space-time boundary element methods for the wave equation

TL;DR

This paper develops stable space-time boundary element methods for the wave equation by formulating a variational problem as a minimization, ensuring stability and providing an error indicator for adaptive refinement.

Contribution

It introduces a novel mixed saddle point formulation for space-time boundary element methods that guarantees stability and offers an error indicator for the wave equation.

Findings

Method achieves discrete inf-sup stability.

Provides a simple error indicator for adaptivity.

Numerical experiments demonstrate applicability to various formulations.

Abstract

In this paper, we recast the variational formulation corresponding to the single layer boundary integral operator $\operatorname{V}$ for the wave equation as a minimization problem in $L^2(\Sigma)$, where $\Sigma := \partial \Omega \times (0,T)$ is the lateral boundary of the space-time domain $Q := \Omega \times (0,T)$. For discretization, the minimization problem is restated as a mixed saddle point formulation. Unique solvability is established by combining conforming nested boundary element spaces for the mixed formulation such that the related bilinear form is discrete inf-sup stable. We analyze under which conditions the discrete inf-sup stability is satisfied, and, moreover, we show that the mixed formulation provides a simple error indicator, which can be used for adaptivity. We present several numerical experiments showing the applicability of the method to different time-domain boundary integral formulations used in the literature.