  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.
