Fritz Kretzschmar, Herbert Egger, Thomas Weiland and I submitted a paper on a space-time discontinuous Galerkin Trefftz method for the time-dependent Maxwell’s equations. Trefftz methods require the basis functions to fulfill the underlying PDEs in an exact sense. Consequently, vectorial basis functions in a space-time setting have to be considered. It is shown that we obtain a largely reduced number of degrees of freedom.
EDIT: The paper was published in the SIAM Journal on Scientific Computing (SISC) 37(5).
The abstract reads
We consider the discretization of electromagnetic wave propagation problems by a discontinuous Galerkin method based on Trefftz polynomials. This method fits into an abstract framework for space-time discontinuous Galerkin methods for which we can prove consistency, stability, and energy dissipation without the need to completely specify the approximation spaces in detail. Any method of such a general form results in an implicit time-stepping scheme with some basic stability properties. For the local approximation on each space-time element, we then consider Trefftz polynomials, i.e., the subspace of polynomials that satisfy Maxwell’s equations exactly on the respective element. We present an explicit construction of a basis for the local Trefftz spaces in two and three dimensions and summarize some of their basic properties. Using local properties of the Trefftz polynomials, we can establish the well-posedness of the resulting discontinuous Galerkin Trefftz method. Consistency, stability, and energy dissipation then follow immediately from the results about the abstract framework. The method proposed in this paper therefore shares many of the advantages of more standard discontinuous Galerkin methods, while at the same time, it yields a substantial reduction in the number of degrees of freedom and the cost for assembling. These benefits and the spectral convergence of the scheme are demonstrated in numerical tests.
Please find the preprint on arXiv.
For more information on the DG Trefftz method please see also our previous paper, which includes the exact treatment of inhomogeneous materials in partially filled cells as immersed boundaries and this link regarding transparent boundary conditions.
UPDATE May, 28, 2015: The paper was accepted.