A local space–time discontinuous finite element method

We present a locally enriched space–time finite element method for solving hyperbolic problems with discontinuities. The space–time formulation is coupled to a standard semidiscrete finite element method. The discontinuities are captured by a space–time version of the extended finite element method, which treats arbitrary moving discontinuities. Since the discontinuities are local, the enriched space–time method is only needed around the discontinuities, which provides significant computational savings. The coupling is implemented through a weak enforcement of the continuity of the flux between the space–time and semidiscrete domains in a manner similar to discontinuous Galerkin methods. The method is illustrated through one-dimensional problems. It displays remarkable accuracy in capturing moving discontinuities, both in their amplitude and speed.