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.

/pdf/a-space-time-trefftz-discontinuous-galerkin-method-for-the-54wn6m38o2.pdf

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

Solving nonlinear direct and inverse problems of stationary heat transfer by using Trefftz functions

A new space-time discontinuous Galerkin (dG) method utilizing special Trefftz polynomial basis functions is proposed and fully analyzed for the scalar wave equation in a second order formulation. The dG method considered is motivated by the class of interior penalty dG methods, as well as by the classical work of Hughes and Hulbert [Comput. Methods Appl. Mech. Engrg., 66 (1988), pp. 339--363; Comput. Methods Appl. Mech. Engrg., 84 (1990), pp. 327--348]. The choice of the penalty terms included in the bilinear form is essential for both the theoretical analysis and for the practical behavior of the method for the case of lowest order basis functions. A best approximation result is proven for this new space-time dG method with Trefftz-type basis functions. Rates of convergence are proved in any dimension and verified numerically in spatial dimensions d = 1 and d = 2. Numerical experiments highlight the effectivness of the Trefftz method in problems with energy at high frequencies.

https://lra.le.ac.uk/bitstream/2381/39785/5/16m1065744.pdf

A Trefftz polynomial space-time discontinuous Galerkin method for the second order wave equation

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. (2014) and of Monk & Richter (2005). 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.

https://www.reading.ac.uk/nmsruntime/saveasdialog.aspx?lID=105745&sID=90309

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

Direct and inverse heat transfer in non-contacting face seals

The paper demonstrates a specific power-series-expansion technique to solve approximately the two-dimensional wave equation. As solving functions (Trefftz functions) so-called wave polynomials are used. The presented method is useful for a finite body of certain shape geometry. Recurrent formulas for the wave polynomials and their derivatives are obtained in the Cartesian and polar coordinate system. The accuracy of the method is discussed and some examples are shown.

/pdf/solution-of-the-two-dimensional-wave-equation-by-using-wave-5ab1tabajr.pdf

Solution of the two-dimensional wave equation by using wave polynomials

https://onlinelibrary.wiley.com/doi/pdf/10.1002/pamm.200410334

Solution of the three-dimension wave equation by using wave polynomials

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Solution of the two-dimensional wave equation by using wave polynomials

Solution of the three-dimension wave equation by using wave polynomials