  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.
