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.

The abstract reads

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.

For more information on the DG Trefftz method please see the posts tagged Trefftz DG.

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).

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.

Group of the Week (V): MONO

As the regular follower of the Group of the Week series that you are, you have of course noticed that I have posted two groups within the same week. I meant to post the previous one on Beck even before Christmas but something annoying like work got in the way.

Anyway, here we go with a great discovery:
MONO, a post-rock/experimental rock group from Japan. When I saw the announcement of a concert in Zurich, I remembered that a few years back I read a review of their album For My Parents in the (german) music review page Abgehört. I also remember that I liked it at the time but probably did not listen to it very intensely. Now, I went back to listen to their music. They have a new double feature album out The Last Dawn and Rays Of Darkness, which received another outstanding review on Abgehört but I think you just need to listen to it yourself, which you can do (and afterwards buy) on Bandcamp. The Hymn to the Immortal Wind is maybe one of the greatest albums I’ve listened to (and the others aren’t really worse).

They played a great concert with a full orchestra supporting them in NY. Other concert videos are worthwhile too, and I am sure that the show in Zurich will be no worse. See you there 🙂

As usual you can listen to it on Spotify:

Group of the Week (IV): Beck

Probably everybody those who are older than 30 years of age know Beck‘s iconic song Loser. Still today I think it has one of the best music videos ever made.

Apparently, Beck has mostly worked as a producer/song writer in the last years, notably (imo) on Charlotte Gainsbourg‘s album IRM.

Last year Beck was back with a new album under his own name. It’s called Morning Phase. As usual you can listen to it on Spotify: