Discontinuous Galerkin Methods with Trefftz Approximation

by F Kretzschmar, S M Schnepp, I Tsukerman, T Weiland
Abstract:
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.
Reference:
Discontinuous Galerkin Methods with Trefftz Approximation (F Kretzschmar, S M Schnepp, I Tsukerman, T Weiland), In Journal of Computational and Applied Mathematics, volume 270, 2014.
Bibtex Entry:
@article{Kretzschmar:2013wt,
author = {Kretzschmar, F and Schnepp, S M and Tsukerman, I and Weiland, T},
title = {{Discontinuous Galerkin Methods with Trefftz Approximation}},
journal = {Journal of Computational and Applied Mathematics},
year = 2014,
volume = 270,
pages = {211-222},
abstract = {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.},
doi = {10.1016/j.cam.2014.01.033},
url = {http://www.sciencedirect.com/science/article/pii/S0377042714000612}
}

This entry was posted by . Bookmark the permalink.