Non-dissipative space-time $hp$-discontinuous Galerkin method for the time-dependent Maxwell Equations

A finite element method for the solution of the time-dependent Maxwell equations in mixed form is presented. The method allows for local $hp$-refinement in space and in time. To this end, a space-time Galerkin approach is employed. In contrast to the space-time DG method introduced in \cite{vegt_space_2002} test and trial space do not coincide. This allows for obtaining a non-dissipative method. In order to obtain an efficient implementation, a hierarchical tensor product basis in space and time is proposed. In particular it allows to evaluate the local residual with a complexity of $\mathcal{O}(p^4)$ and $\mathcal{O}(p^5)$ for affine and non-affine elements, respectively.