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In this paper, we recast the variational formulation corresponding to thesingle layer boundary integral operator $\operatorname{V}$ for the waveequation 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 restatedas a mixed saddle point formulation. Unique solvability is established bycombining conforming nested boundary element spaces for the mixed formulationsuch that the related bilinear form is discrete inf-sup stable. We analyzeunder 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 experimentsshowing the applicability of the method to different time-domain boundaryintegral formulations used in the literature.

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