Discontinuous Galerkin methods with transient hp adaptation

A discontinuous Galerkin method (DGM) for Maxwell's equations in time domain and dedicated techniques for adaptive mesh refinement are presented. Since the DGM is a finite element-type method, it offers two refinement mechanisms: the manipulation of the local mesh step size (h adaptation) and the adaptation of the local approximation order (p adaptation). For both cases, a new approximation is obtained by means of projections between finite element spaces. The projection operators introduced are optimal with respect to the projection error. A reliable estimator for the local smoothness of the solution is presented, which forms the basis for the hp decision, i.e., the choice of the type of adaptation to be performed. The stability and efficiency of the adaptive method are demonstrated, allowing for performing transient mesh refinement, i.e., the continuous adaptation of the mesh according to the current situation.