A Survey of Trefftz Methods for the Helmholtz Equation

Abstract

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.

The original version of this chapter was revised. An erratum to this chapter can be found at DOI 10.1007/978-3-319-41640-3_14.

An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-3-319-41640-3_14

Appendix: Condition Numbers of Plane Wave Mass Matrices

Given a wave number k > 0 and $p \in \mathbb{N}$ distinct unit vectors $\mathbf{d}_{\ell} \in \mathbb{R}^{n}$, ℓ = 1, …, p, and a domain $K \subset \mathbb{R}^{n}$ with barycentre x _K , the symmetric positive definite plane wave element mass matrix M _K on K is defined as

$$\displaystyle{ \mathbf{M}_{K}:= \left (\int \nolimits _{K}e^{\mathrm{i}k\mathbf{d}_{\ell}\cdot (\mathbf{x}-\mathbf{x}_{K})} \cdot e^{-\mathrm{i}k\mathbf{d}_{m}\cdot (\mathbf{x}-\mathbf{x}_{K})}\,\mathrm{d}V \right )_{\ell,m=1}^{p}. }$$

For n = 2 we computed spectral condition numbers of M _K for equi-spaced directions d _ℓ  = (cos(2π ℓ∕p), sin(2π ℓ∕p)), ℓ = 0, …, p − 1. For n = 3 we choose the directions d _ℓ as the “minimum norm points” according to Sloan and Womersley [103, 116]. These points are indexed by a level $q \in \mathbb{N}$ and p = (q + 1)². The spectral condition numbers are plotted in Fig. 1 for n = 2, K = (−1, 1)², and Fig. 2 for n = 3, K = (−1, 1)³. They have been computed with MATLAB using the high-precision arithmetic (200 decimal digits) provided by the Advanpix Multi-Precision Toolbox.³

Fig. 1

Condition numbers of element mass matrices on the square (−1, 1)²

Fig. 2

Condition numbers of element mass matrices on the cube (−1, 1)³