Trefftz Discontinuous Galerkin Research Articles

Trefftz discontinuous Galerkin discretization for the Stokes problem

We introduce a new discretization based on a polynomial Trefftz-DG method for solving the Stokes equations. Discrete solutions of this method fulfill the Stokes equations pointwise within each element and yield element-wise divergence-free solutions. Compared to standard DG methods, a strong reduction of the degrees of freedom is achieved, especially for higher polynomial degrees. In addition, in contrast to many other Trefftz-DG methods, our approach allows us to easily incorporate inhomogeneous right-hand sides (driving forces) by using the concept of the embedded Trefftz-DG method. On top of a detailed a priori error analysis, we further compare our approach to other (hybrid) discontinuous Galerkin Stokes discretizations and present numerical examples.

On polynomial Trefftz spaces for the linear time-dependent Schrödinger equation

We study the approximation properties of complex-valued polynomial Trefftz spaces for the (d+1)-dimensional linear time-dependent Schrödinger equation. More precisely, we prove that for the space–time Trefftz discontinuous Galerkin variational formulation proposed by Gómez and Moiola (2022), the same h-convergence rates as for polynomials of degree p in (d+1) variables can be obtained in a mesh-dependent norm by using a space of Trefftz polynomials of anisotropic degree. For such a space, the dimension is equal to that of the space of polynomials of degree 2p in d variables, and bases are easily constructed.

Unfitted Trefftz discontinuous Galerkin methods for elliptic boundary value problems

We propose a new geometrically unfitted finite element method based on discontinuous Trefftz ansatz spaces. Trefftz methods allow for a reduction in the number of degrees of freedom in discontinuous Galerkin methods, thereby, the costs for solving arising linear systems significantly. This work shows that they are also an excellent way to reduce the number of degrees of freedom in an unfitted setting. We present a unified analysis of a class of geometrically unfitted discontinuous Galerkin methods with different stabilisation mechanisms to deal with small cuts between the geometry and the mesh. We cover stability and derive a-priori error bounds, including errors arising from geometry approximation for the class of discretisations for a model Poisson problem in a unified manner. The analysis covers Trefftz and full polynomial ansatz spaces, alike. Numerical examples validate the theoretical findings and demonstrate the potential of the approach.

Embedded Trefftz discontinuous Galerkin methods

AbstractIn 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.

A Space-Time Trefftz Discontinuous Galerkin Method for the Linear Schrödinger Equation

A space-time Trefftz discontinuous Galerkin method for the Schr\"odinger equation with piecewise-constant potential is proposed and analyzed. Following the spirit of Trefftz methods, trial and test spaces are spanned by non-polynomial complex wave functions that satisfy the Schro\"odinger equation locally on each element of the space-time mesh. This allows for a significant reduction in the number of degrees of freedom in comparison with full polynomial spaces. We prove well-posedness and stability of the method, and, for the one- and two- dimensional cases, optimal, high-order, h-convergence error estimates in a skeleton norm. Some numerical experiments validate the theoretical results presented.

Space–time discontinuous Galerkin approximation of acoustic waves with point singularities

AbstractWe develop a convergence theory of space–time discretizations for the linear, second-order wave equation in polygonal domains $\varOmega \subset{\mathbb R}^2$, possibly occupied by piecewise homogeneous media with different propagation speeds. Building on an unconditionally stable space–time DG formulation developed in Moiola & Perugia (2018, A space–time Trefftz discontinuous Galerkin method for the acoustic wave equation in first-order formulation. Numer. Math., 138, 389–435), we (a) prove optimal convergence rates for the space–time scheme with local isotropic corner mesh refinement on the spatial domain, and (b) demonstrate numerically optimal convergence rates of a suitable sparse space–time version of the DG scheme. The latter scheme is based on the so-called combination formula, in conjunction with a family of anisotropic space–time DG discretizations. It results in optimal-order convergent schemes, also in domains with corners, with a number of degrees of freedom that scales essentially like the DG solution of one stationary elliptic problem in $\varOmega $ on the finest spatial grid. Numerical experiments for both smooth and singular solutions support convergence rate optimality on spatially refined meshes of the full and sparse space–time DG schemes.

Trefftz discontinuous Galerkin basis functions for a class of Friedrichs systems coming from linear transport

The Trefftz discontinuous Galerkin (TDG) method provides natural well-balanced (WB)and asymptotic preserving (AP) discretization, since exact solutions are used locally in the basis functions. However, one difficult point may be the construction of such solutions, which is a necessary first step in order to apply the TDG scheme. This work deals with the construction of solutions to Friedrichs systems with relaxation with application to the spherical harmonics approximation of the transport equation (the so-called PN models). Various exponential and polynomial solutions are constructed. Two numerical tests on the P3 model illustrate the good accuracy of the TDG method. They show that the exponential solutions lead to accurate schemes to capture boundary layers on a coarse mesh and that a combination of exponential and polynomial solutions is efficient in a regime with vanishing absorption coefficient.

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

We present a space–time Trefftz discontinuous Galerkin method for approximating the acoustic wave equation semi-explicitly on tent pitched meshes. DG Trefftz methods use discontinuous test and trial functions, which solve the wave equation locally. Tent pitched meshes allow us to solve the equation elementwise, allowing locally optimal advances in time. The method is implemented in NGSolve, solving the space–time elements in parallel, whenever possible. Insights into the implementational details are given, including the case of propagation in heterogeneous media.

A Trefftz-discontinuous Galerkin method for time-harmonic elastic wave problems

In this paper, we develop a plane wave discontinuous Galerkin method combined with local spectral element method for the elastic wave propagation in two and three space dimensions. We derive the error estimates of the approximation solutions in the mesh-dependent norm and the mesh-independent norm. Some dependence of the error bounds on the orders q of local spectral elements and the number p of plane wave propagation directions is given. Numerical results assess the validity of the theoretical results and indicate that the resulting approximate solutions generated by the PWDG–LSFE possess high accuracy.

A Trefftz Discontinuous Galerkin method for time-harmonic waves with a generalized impedance boundary condition

ABSTRACTWe show how a Trefftz Discontinuous Galerkin (TDG) method for the displacement form of the Helmholtz equation can be used to approximate problems having a generalized impedance boundary condition (GIBC) involving surface derivatives of the solution. Such boundary conditions arise naturally when modeling scattering from a scatterer with a thin coating. The thin coating can then be approximated by a GIBC. A second place GIBCs arise is as higher order absorbing boundary conditions. This paper also covers both cases. Because the TDG scheme has discontinuous elements, we propose to couple it to a surface discretization of the GIBC using continuous finite elements. We prove convergence of the resulting scheme and demonstrate it with two numerical examples.

Adaptive refinement for hp–Version Trefftz discontinuous Galerkin methods for the homogeneous Helmholtz problem

In this article we develop an hp-adaptive refinement procedure for Trefftz discontinuous Galerkin methods applied to the homogeneous Helmholtz problem. Our approach combines not only mesh subdivision (h-refinement) and local basis enrichment (p-refinement), but also incorporates local directional adaptivity, whereby the elementwise plane wave basis is aligned with the dominant scattering direction. Numerical experiments based on employing an empirical a posteriori error indicator clearly highlight the efficiency of the proposed approach for various examples.

Trefftz Discontinuous Galerkin Method for Friedrichs Systems with Linear Relaxation: Application to the P 1 Model

Abstract This work deals with the first Trefftz Discontinuous Galerkin (TDG) scheme for a model problem of transport with relaxation. The model problem is written as a P N {P_{N}} or S N {S_{N}} model, and we study in more details the P 1 {P_{1}} model in dimension 1 and 2. We show that the TDG method provides natural well-balanced and asymptotic preserving discretization since exact solutions are used locally in the basis functions. High-order convergence with respect to the mesh size in two dimensions is proved together with the asymptotic property for P 1 {P_{1}} model in dimension one. Numerical results in dimensions 1 and 2 illustrate the theoretical properties.

On wave based weak Trefftz discontinuous Galerkin approach for medium-frequency heterogeneous Helmholtz problem

Medium frequency heterogeneous Helmholtz problem gives rise to severe numerical challenge to Finite Element Method (FEM). An extended Variational Theory of Complex Rays (VTCR) is developed for solving this problem. Since VTCR is kind of Trefftz method, analytic solutions of governing equation need to be known a priori. But this requirement limits the application of VTCR to some specific forms of governing equations. Recently, a new computational method named weak Trefftz discontinuous Galerkin method (WTDG) is proposed to enable hybrid Finite Element method (FEM)/ Variational Theory of Complex Rays (VTCR) strategies to be developed easily in acoustics and vibrations. This hybrid method combines the advantages of FEM and VTCR to solve efficiently low frequency and medium frequency problems. This paper proposes a wave based WTDG for heterogeneous Helmholtz problem. Without knowing analytic solution of governing equation, WTDG method provides a simple and general resolution strategy for medium frequency heterogeneous Helmholtz problem.

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. (2016, IMA J. Numer. Anal., 36, 1599-1635). 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.

Extended variational theory of complex rays in heterogeneous Helmholtz problem

In the past years, a numerical technique method called Variational Theory of Complex Rays (VTCR) has been developed for vibration problems in medium frequency. It is a Trefftz Discontinuous Galerkin method which uses plane wave functions as shape functions. However this method is only well developed in homogeneous case. In this paper, VTCR is extended to the heterogeneous Helmholtz problem by creating a new base of shape functions. Numerical examples give a scope of the performances of such an extension of VTCR.

Hybrid Finite Element Method and Variational Theory of Complex Rays for Helmholtz Problems

In this paper, we consider the Weak Trefftz Discontinuous Galerkin (WTDG) method, which enables one to use at the same time the Finite Element Method (FEM) or Variational Theory of Complex Rays (VTCR) discretizations (polynoms and waves), for vibration problems. It has already been developed such that the FEM and the VTCR can be used in different adjacent subdomains in the same problem. Here, it is revisited and extended in order to allow one to use the two discretizations in the same subdomain, at the same time. Numerical examples illustrate the performances of such an approach.

Numerical simulation of wave propagation in inhomogeneous media using Generalized Plane Waves

The Trefftz Discontinuous Galerkin (TDG) method is a technique for approximating the Helmholtz equation (or other linear wave equations) using piecewise defined local solutions of the equation to approximate the global solution. When coefficients in the equation (for example, the refractive index) are piecewise constant it is common to use plane waves on each element. However when the coefficients are smooth functions of position, plane waves are no longer directly applicable. In this paper we show how Generalized Plane Waves (GPWs) can be used in a modified TDG scheme to approximate the solution for piecewise smooth coefficients. GPWs are approximate solutions to the equation that reduce to plane waves when the medium through which the wave propagates is constant. We shall show how to modify the TDG sesquilinear form to allow us to prove convergence of the GPW based version. The new scheme retains the high order convergence of the original TDG scheme (when the solution is smooth) and also retains the same number of degrees of freedom per element (corresponding to the directions of the GPWs). Unfortunately it looses the advantage that only skeleton integrals need to be performed. Besides proving convergence, we provide numerical examples to test our theory.

A symmetric Trefftz-DG formulation based on a local boundary element method for the solution of the Helmholtz equation

A general symmetric Trefftz Discontinuous Galerkin method is built for solving the Helmholtz equation with piecewise constant coefficients. The construction of the corresponding local solutions to the Helmholtz equation is based on a boundary element method. A series of numerical experiments displays an excellent stability of the method relatively to the penalty parameters, and more importantly its outstanding ability to reduce the instabilities known as the “pollution effect” in the literature on numerical simulations of long-range wave propagation.

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

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., and of Monk and Richter. 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.

Plane Wave Discontinuous Galerkin Methods: Exponential Convergence of the $$hp$$ h p -Version

We consider the two-dimensional Helmholtz equation with constant coefficients on a domain with piecewise analytic boundary, modelling the scattering of acoustic waves at a sound-soft obstacle. Our discretisation relies on the Trefftz-discontinuous Galerkin approach with plane wave basis functions on meshes with very general element shapes, geometrically graded towards domain corners. We prove exponential convergence of the discrete solution in terms of number of unknowns.