# Book Chapter on the Power of Trefftz Approximations published

Fritz Kretzschmar, Herbert Egger, Farzad Ahmadi, Nabil Nowak, Vadim A. Markel, Igor Tsukerman and myself contributed a chapter entitled

The Power of Trefftz Approximations: Finite Difference, Boundary Difference and Discontinuous Galerkin Methods; Nonreflecting Conditions and Non-Asymptotic Homogenization

to the book Finite Difference Methods,Theory and Applications Volume 9045 of the series Lecture Notes in Computer ScienceThe abstract reads

In problems of mathematical physics, Trefftz approximations by definition involve functions that satisfy the differential equation of the problem. The power and versatility of such approximations is illustrated with an overview of a number of application areas: (i) finite difference Trefftz schemes of arbitrarily high order; (ii) boundary difference Trefftz methods analogous to boundary integral equations but completely singularity-free; (iii) Discontinuous Galerkin (DG) Trefftz methods for Maxwell’s electrodynamics; (iv) numerical and analytical nonreflecting Trefftz boundary conditions; (v) non-asymptotic homogenization of electromagnetic and photonic metamaterials.

# Paper on the A priori error analysis of space-time Trefftz discontinuous Galerkin methods for wave problems

Fritz Kretzschmar, Andrea Moiola, Ilaria Perugia and I prepared a paper on the a priori error analysis of space-time Trefftz discontinuous Galerkin methods for wave problems.

We present and analyse a space-time discontinuous Galerkin method for wave propagation problems. The special feature of the scheme is that it is a Trefftz method, namely that trial and test functions are solution of the partial differential equation to be discretised in each element of the (space-time) mesh. The method considered is a modification of the discontinuous Galerkin schemes of Kretzschmar et al., and of Monk and Richter. For Maxwell’s equations in one space dimension, we prove stability of the method, quasi-optimality, best approximation estimates for polynomial Trefftz spaces and (fully explicit) error bounds with high order in the meshwidth and in the polynomial degree. The analysis framework also applies to scalar wave problems and Maxwell’s equations in higher space dimensions. Some numerical experiments demonstrate the theoretical results proved and the faster convergence compared to the non-Trefftz version of the scheme.

Please find the preprint on arXiv.

EDIT Oct/28/2015: The paper got accepted by IMA Journal of Numerical Analysis
EDIT Dec/18/2015: Please find the final paper online.

# Paper on A Space-Time Discontinuous Galerkin Trefftz Method for the Time-Dependent Maxwell’s Equations

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).

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.

# Paper on Transparent Boundary Conditions in a Discontinuous Galerkin Trefftz method

Fritz Kretzschmar, Herbert Egger, Igor Tsukerman, Thomas Weiland and I submitted a paper on transparent boundary conditions in the Discontinuous Galerkin Trefftz method.

EDIT: The paper was published in Applied Mathematics and Computation
Volume 267, Pages 42–55

The modeling and simulation of electromagnetic wave propagations is often accompanied by a restriction to bounded domains and the introduction of artificial boundary conditions which should be chosen in order to minimize parasitic reflections. In this paper, we investigate a new type of transparent boundary condition and its implementation in a Discontinuous Galerkin Trefftz Finite Element Method. The choice of a particular set of basis functions allows us to split the electromagnetic field into components with a 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 in comparison with commonly used absorbing boundary conditions. In our numerical tests, we observe spectral convergence in the L2 norm and a dissipative behavior for which we provide a theoretical explanation.

# Student Paper Competition Award for Fritz Kretzschmar

Fritz Kretzschmar received a Student Paper Competition Award for his contribution entitled

The Discontinuous Galerkin Trefftz Method

at the 12th International Workshop on Finite Elements for Microwave Engineering (FEM2014). Coauthors are Igor Tsukerman, Thomas Weiland and myself. Congratulations!

If you are interested in more information on the time-domain DG Trefftz method have a look at our recent paper in JCAM and/or contact Fritz or me.

# Paper on Discontinuous Galerkin methods with Trefftz approximations published in JCAM

Fritz Kretzschmar, Igor Tsukerman and I have a new paper in Journal of Computational Applied Mathematics (JCAM). The paper abstract reads

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.