Discontinuous Galerkin Methods with Trefftz Approximation

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.

14 pages, 12 figures, preprint submitted at J Comput Phys

Cited by in corpus (10)

• Trefftz discontinuous Galerkin methods on unstructured meshes for the wave equation
• A Survey of Trefftz Methods for the Helmholtz Equation
• A space-time Trefftz discontinuous Galerkin method for the acoustic wave equation in first-order formulation
• A priori error analysis of space-time Trefftz discontinuous Galerkin methods for wave problems
• von Neumann Stability Analysis of Globally Constraint-Preserving DGTD and PNPM Schemes for the Maxwell Equations using Multidimensional Riemann Solvers
• A space-time quasi-Trefftz DG method for the wave equation with piecewise-smooth coefficients
• Approximately well-balanced Discontinuous Galerkin methods using bases enriched with Physics-Informed Neural Networks
• The FLAME-slab method for electromagnetic wave scattering in aperiodic slabs
• Inf-sup stable space-time Local Discontinuous Galerkin method for the heat equation
• Trefftz Functions for Nonlocal Electrostatics