Discontinuous Galerkin Methods with Trefftz Approximation
This provides a novel high-order method for wave propagation problems that inherently includes high-order time integration, distinguishing it from methods with only spatial high-order approximation.
The paper introduces a Discontinuous Galerkin method using space-time Trefftz basis functions for wave propagation, achieving spectral convergence in the L2-norm across the entire space-time domain.
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.