A priorierror analysis of space–time Trefftz discontinuous Galerkin methods for wave problems

Abstract

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. [25] and of Monk and Richter [28]. 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.

Tent pitching and Trefftz-DG method for the acoustic wave equation

“…The use of Trefftz spaces allows to reduce the number of degrees of freedom, as compared to the total degree polynomial spaces, however keeping the same accuracy. Work on Trefftz-DG methods for different wave propagation problems includes [3,4,6,7,15,16,19]. In [3], a Trefftz-DG method in space-time for the second order wave equation is presented, proving h-convergence in 1, 2 and 3 space dimensions, as well as hp-convergence, along with exponential convergence for analytic solutions in 1 space dimension.…”

A space–time Trefftz discontinuous Galerkin method for the acoustic wave equation in first-order formulation

“…The Trefftz formulation for Maxwell's equations of [10,11,23,25] corresponds to the "unpenalised" version of that one proposed here (the numerical experiments in [24, §7.5] show that the numerical error depends very mildly on the penalisation parameters).We first describe the IBVP under consideration in §2, the assumptions on the mesh in §3 and the Trefftz-DG formulation in §4. Following the thread of [18,24], in §5.2 and §5.3 we prove that the scheme is well-posed, quasi-optimal, dissipative (quantifying dissipation using the jumps of the discrete solution), and derive error estimates for some traces of the solution on the mesh skeleton. In §5.4 we investigate how to control the Trefftz-DG error in a mesh-independent norm: after setting up a general duality framework in §5.4.1, we prove error bounds in L 2 (Q) norm (Q being the space-time computational domain) under some restrictive assumptions on the mesh in §5.4.2, and in a weaker Sobolev norm in §5.4.3 under different assumptions.The analysis carried out in §5 holds for any choice of discrete Trefftz spaces.…”

“…For both discrete spaces we introduce simple bases and prove approximation estimates, which lead to fully explicit, high-order (in the meshwidth h), optimal-in-h convergence estimates for the Trefftz-DG method; see Theorems 6.8 and 6.19. Estimates ensuring convergence with respect to the polynomial degree p, such as those proved in [24, §5.3.2] for one space dimension, are still elusive in the general case; the same situation occurs in [3].The analysis differs from that of [24] in several respects: we consider higher-dimensional problems (which is the most fundamental difference), space-like element faces not necessarily perpendicular to the time axis, error bounds in mesh-independent norms other than L 2 (since bounds in L 2 norm do not seem possible in this generality), we use different techniques to prove approximation properties of Trefftz polynomials (restricted to h-convergence only). We expect that all results presented here, except possibly those of §5.4.3 on error bounds in mesh-independent norm in the presence of time-like faces, can be extended to the case of Maxwell's equations in three space dimensions in a straightforward way.Comparing against the Trefftz scheme of [3] which is of interior penalty type, our error analysis does not use inverse estimates for polynomials, thus the analysis holds for any discrete Trefftz space (including non-polynomial ones, cf.…”

“…Remark 6.15) and the numerical flux parameters in the definition of the formulation are more easily determined (i.e. no parameter has to be "large enough").One of the strengths of the Trefftz-DG method compared to non-Trefftz schemes is the much better asymptotic behaviour in terms of accuracy per number of degrees of freedom; this has already been described in details and demonstrated numerically in [24]. More precisely, for a problem in n space dimensions, the Trefftz approach allows to reduce from O(p n+1 ) to O(p n ) the dimension of local space-time approximating spaces with effective h-approximation…”

Space-Time Discontinuous Galerkin Discretizations for Linear First-Order Hyperbolic Evolution Systems

“…Both methods are implicit in every time slab, and only Dirichlet traces are used for the hybrid coupling. Space-time (Trefftz) discontinuous Galerkin methods for wave problems are analyzed in [8,23].…”