Thomas Weiland

Sascha M. Schnepp, Thomas Weiland

A framework for performing dynamic mesh adaptation with the discontinuous Galerkin method (DGM) is presented. Adaptations include modifications of the local mesh step size (h-adaptation) and the local degree of the approximating polynomials (p-adaptation) as well as their combination. The computation of the approximation within locally adapted elements is based on projections between finite element spaces (FES), which are shown to preserve an upper limit of the electromagnetic energy. The formulation supports high level hanging nodes and applies precomputation of surface integrals for increasing computational efficiency. Error and smoothness estimates based on interface jumps are presented and applied to the fully hp-adaptive simulation of two examples in one-dimensional space. A full wave simulation of electromagnetic scattering form a radar reflector demonstrates the applicability to large scale problems in three-dimensional space.

Fritz Kretzschmar, Sascha Schnepp, Igor Tsukerman et al.

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 $\Lebesgue_2$-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.

Herbert Egger, Fritz Kretzschmar, Sascha M. Schnepp et al.

The modeling and simulation of electromagnetic wave propagation is often accompanied by a restriction to bounded domains which requires the introduction of artificial boundaries. The corresponding boundary conditions should be chosen in order to minimize parasitic reflections. In this paper, we investigate a new type of transparent boundary condition for a discontinuous Galerkin Trefftz finite element method. The choice of a particular basis consisting of polynomial plane waves allows us to split the electromagnetic field into components with a well specified direction of propagation. The reflections at the artificial boundaries are then reduced by penalizing components of the field incoming into the space-time domain of interest. We formally introduce this concept, discuss its realization within the discontinuous Galerkin framework, and demonstrate the performance of the resulting approximations by numerical tests. A comparison with first order absorbing boundary conditions, that are frequently used in practice, is made. For a proper choice of basis functions, we observe spectral convergence in our numerical test and an overall dissipative behavior for which we also give some theoretical explanation.

Martin Lilienthal, Sascha M. Schnepp, Thomas Weiland

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.

Herbert Egger, Fritz Kretzschmar, Sascha M. Schnepp et al.

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.