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

We consider a stable unique continuation problem for the wave equation where the initial data is lacking and the solution is reconstructed using measurements in some subset of the bulk domain. Typically fairly sophisticated space-time methods have been used in previous work to obtain stable and accurate solutions to this reconstruction problem. Here we propose to solve the problem using a standard discontinuous Galerkin method for the temporal discretization and continuous finite elements for the space discretization. Error estimates are established under a geometric control condition. We also investigate two preconditioning strategies which 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.