The unique continuation problem for the wave equation discretized with a high-order space-time nonconforming method

We are interested in solving the unique continuation problem for the wave equation, i.e., we want to reconstruct the solution of the wave equation given its (noised) value in a subset of the computational domain. Homogeneous Dirichlet boundary conditions are imposed, whereas the initial datum is unknown. We discretize this problem using a space-time discontinuous Galerkin method (including hybrid variables in space and in time) and look for the solution corresponding to the saddle-point of a discrete Lagrangian. We establish discrete inf-sup stability and bound the consistency error, leading to a priori estimates on the residual. Our main result proves the convergence of the discrete solution to the exact solution in a shifted energy norm involving weaker Sobolev norms than the standard energy norm for the wave equation. The proof combines the above a priori bound with a conditional stability estimate at the continuous level. Finally, we run numerical simulations to assess the performance of the method in practice. A static condensation procedure is used to eliminate the cell unknowns and reduce the size of the linear system.