We present a novel Discontinuous Galerkin Finite Element Method for wave propagation problems. The method employs space–time Trefftz-type basis functions that satisfy the underlying partial differential equations and the respective interface boundary conditions exactly in an element-wise fashion. The basis functions can be of arbitrary high order, and we demonstrate spectral convergence in the \(L_2\)-norm. In this context, spectral convergence is obtained with respect to the approximation error in the entire space–time domain of interest, i.e. in space and time simultaneously. Formulating the approximation in terms of a space–time Trefftz basis makes high order time integration an inherent property of the method and clearly sets it apart from methods, that employ a high order approximation in space only.
Fritz Kretzschmar, Igor Tsukerman and started working on a novel DG method using a Trefftz-type basis in time-domain. The method is formulated in a space-time FE setting and provides high order approximations in both, space and time, where the high order accuracy in time comes at no extra costs in terms in degrees of freedom. We demonstrated spectral convergence over the full space-time domain.
You can get the preprint on arXiv.
Martin Lilienthal will attend the 17th European Conference on Mathematics for Industry (ECMI) to be held July 23-27 2012 at Lund, Sweden. He will contribute with his talk on a
Non-dissipative space time hp-discontinuous Galerkin method for the time-dependent Maxwell equations
to the mini-symposium organized by Mike Botchev and Wil Schilders.
Fritz Kretzschmar and myself will attend the SCEE conference in Zurich, Switzerland, September 11 – 14, 2012. Fritz will present his work on a novel Discontinuous Galerkin Trefftz-type method for time-domain electromagnetics simulations. I was given the change to contribute with a keynote talk and will summarize my efforts regarding dynamic hp-adaptivity for the Discontinuous Galerkin method.