A Space-Time Discontinuous Galerkin Trefftz Method for Time Dependent Maxwell's Equations

Abstract

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 every 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 discontinuous Galerkin methods based on full polynomial approximation spaces, while at the same time it yields a substantial reduction in the number of degrees of freedom and the cost for assembling. The viability of the method and its spectral convergence are demonstrated in numerical tests.