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

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 :=...

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Main Authors Hoonhout, Daniel, Löscher, Richard, Steinbach, Olaf, Urzúa-Torres, Carolina
Format Journal Article
LanguageEnglish
Published 19.12.2023
Subjects
Computer Science - Numerical Analysis
Mathematics - Numerical Analysis
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Summary: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.
DOI:10.48550/arxiv.2312.12547