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

We introduce a space–time Trefftz discontinuous Galerkin method for the first-order transient acoustic wave equations in arbitrary space dimensions, extending the one-dimensional scheme of Kretzschmar et al. (IMA J Numer Anal 36:1599–1635, 2016). Test and trial discrete functions are space–time piecewise polynomial solutions of the wave equations. We prove well-posedness and a priori error bounds in both skeleton-based and mesh-independent norms. The space–time formulation corresponds to an implicit time-stepping scheme, if posed on meshes partitioned in time slabs, or to an explicit scheme, if posed on “tent-pitched” meshes. We describe two Trefftz polynomial discrete spaces, introduce bases for them and prove optimal, high-order h-convergence bounds

Trefftz discontinuous Galerkin methods on unstructured meshes for the wave equation

We describe and analyse a space-time Trefftz discontinuous Galerkin method for the wave equation. The method is defined for unstructured meshes whose internal faces need not be aligned to the space-time axes. We show that the scheme is well-posed and dissipative, and we prove a priori error bounds for general Trefftz discrete spaces. A concrete discretisation can be obtained using piecewise polynomials that satisfy the wave equation elementwise.Comment: 8 pages, submitted to the XXIV CEDYA / XIV CMA conference, Cadiz 8-12 June 201

On Trefftz and weak Trefftz discontinuous Galerkin approaches for medium-frequency acoustics

International audienceIn this paper, the wave approach called the Variational Theory of Complex Rays (VTCR), which was developed for medium-frequency acoustics and vibrations, is revisited as a discontinuous Galerkin method. Extensions leading to a weak Trefftz constraint are introduced. This weak Trefftz discontinuous Galerkin approach enables hybrid FEM/VTCR strategies to be developed easily, and paves the way for new computational techniques for the resolution of engineering problems. This paper presents some of the fundamental properties of the approach, which is illustrated by several numerical examples

Embedded Trefftz discontinuous Galerkin methods

In Trefftz discontinuous Galerkin methods a partial differential equation is discretized using discontinuous shape functions that are chosen to be elementwise in the kernel of the corresponding differential operator. We propose a new variant, the embedded Trefftz discontinuous Galerkin method, which is the Galerkin projection of an underlying discontinuous Galerkin method onto a subspace of Trefftz-type. The subspace can be described in a very general way and to obtain it no Trefftz functions have to be calculated explicitly, instead the corresponding embedding operator is constructed. In the simplest cases the method recovers established Trefftz discontinuous Galerkin methods. But the approach allows to conveniently extend to general cases, including inhomogeneous sources and non-constant coefficient differential operators. We introduce the method, discuss implementational aspects and explore its potential on a set of standard PDE problems. Compared to standard discontinuous Galerkin methods we observe a severe reduction of the globally coupled unknowns in all considered cases, reducing the corresponding computing time significantly. Moreover, for the Helmholtz problem we even observe an improved accuracy similar to Trefftz discontinuous Galerkin methods based on plane waves.Comment: 25 pages, 14 figures, 1 tabl

Enriched discrete spaces for time domain wave equations

The second order linear wave equation is simple in representation but its numerical approximation is challenging, especially when the system contains waves of high frequencies. While 10 grid points per wavelength is regarded as the rule of thumb to achieve tolerable approximation with the standard numerical approach, high resolution or high grid density is often required at high frequency which is often computationally demanding. As a contribution to tackling this problem, we consider in this thesis the discretization of the problem in the framework of the space-time discontinuous Galerkin (DG) method while investigating the solution in a finite dimensional space whose building blocks are waves themselves. The motivation for this approach is to reduce the number of degrees of freedom per wavelength as well as to introduce some analytical features of the problem into its numerical approximation. The developed space-time DG method is able to accommodate any polynomial bases. However, the Trefftz based space-time method proves to be efficient even for a system operating at high frequency. Comparison with polynomial spaces of total degree shows that equivalent orders of convergence are obtainable with fewer degrees of freedom. Moreover, the implementation of the Trefftz based method is cheaper as integration is restricted to the space-time mesh skeleton. We also extend our technique to a more complicated wave problem called the telegraph equation or the damped wave equation. The construction of the Trefftz space for this problem is not trivial. However, the exibility of the DG method enables us to use a special technique of propagating polynomial initial data using a wave-like solution (analytical) formula which gives us the required wave-like local solutions for the construction of the space. This thesis contains important a priori analysis as well as the convergence analysis for the developed space-time method, and extensive numerical experiments

A space-time quasi-Trefftz DG method for the wave equation with piecewise-smooth coefficients

Trefftz methods are high-order Galerkin schemes in which all discrete functions are elementwise solution of the PDE to be approximated. They are viable only when the PDE is linear and its coefficients are piecewise constant. We introduce a 'quasi-Trefftz' discontinuous Galerkin method for the discretisation of the acoustic wave equation with piecewise-smooth wavespeed: the discrete functions are elementwise approximate PDE solutions. We show that the new discretisation enjoys the same excellent approximation properties as the classical Trefftz one, and prove stability and high-order convergence of the DG scheme. We introduce polynomial basis functions for the new discrete spaces and describe a simple algorithm to compute them. The technique we propose is inspired by the generalised plane waves previously developed for time-harmonic problems with variable coefficients; it turns out that in the case of the time-domain wave equation under consideration the quasi-Trefftz approach allows for polynomial basis functions.Comment: 25 pages, 9 figure

A survey of Trefftz methods for the Helmholtz equation

Trefftz methods are finite element-type schemes whose test and trial functions are (locally) solutions of the targeted differential equation. They are particularly popular for time-harmonic wave problems, as their trial spaces contain oscillating basis functions and may achieve better approximation properties than classical piecewise-polynomial spaces. We review the construction and properties of several Trefftz variational formulations developed for the Helmholtz equation, including least squares, discontinuous Galerkin, ultra weak variational formulation, variational theory of complex rays and wave based methods. The most common discrete Trefftz spaces used for this equation employ generalised harmonic polynomials (circular and spherical waves), plane and evanescent waves, fundamental solutions and multipoles as basis functions; we describe theoretical and computational aspects of these spaces, focusing in particular on their approximation properties. One of the most promising, but not yet well developed, features of Trefftz methods is the use of adaptivity in the choice of the propagation directions for the basis functions. The main difficulties encountered in the implementation are the assembly and the ill-conditioning of linear systems, we briefly survey some strategies that have been proposed to cope with these problems.Comment: 41 pages, 2 figures, to appear as a chapter in Springer Lecture Notes in Computational Science and Engineering. Differences from v1: added a few sentences in Sections 2.1, 2.2.2 and 2.3.1; inserted small correction