Unique continuation for the wave equation based on a discontinuous Galerkin time discretization

We consider a stable unique continuation problem for the wave equation that has been discretized so far using fairly sophisticated space-time methods. Here, we propose to solve this problem using a standard discontinuous Galerkin method for the temporal discretization. Error estimates are established under a geometric control condition. We also investigate two preconditioning strategies that can be used to solve the arising globally coupled space-time system by means of simple time-stepping procedures. Our numerical experiments test the performance of these strategies and highlight the importance of the geometric control condition for reconstructing the solution beyond the data domain.