In this paper we are concerned with Trefftz discretizations of the time-dependent linear wave equation in anisotropic media in arbitrary space dimensional domains $\Omega \subset \mathbb{R}^d~ (d\in \mathbb{N})$. We propose two variants of the Trefftz DG method, define novel plane wave basis functions based on rigorous choices of scaling transformations and coordinate transformations, and prove that the corresponding approximate solutions possess optimal-order error estimates with respect to the meshwidth $h$ and the condition number of the coefficient matrices, respectively. Besides, we propose the global Trefftz DG method combined with local DG methods to solve the time-dependent linear nonhomogeneous wave equation in anisotropic media. In particular, the error analysis holds for the (nonhomogeneous) Dirichlet, Neumann, and mixed boundary conditions from the original PDEs. Furthermore, a strategy to discretize the model in heterogeneous media is proposed. The numerical results verify the validity of the theoretical results, and show that the resulting approximate solutions possess high accuracy.

翻译：在本文中,我们关注在任意空间空间维域中的厌食介质中时间依赖线性波方程式的Trefftz离散问题 $\ Omega\ subset \ mathbb{R ⁇ d~ (d\ in\ mathb{N})$。我们提出了Trefftz DG方法的两个变式,根据严格的缩放变换和协调变异,定义新的平面波基功能,并证明相应的近似解决办法分别拥有关于网状美元和系数矩阵条件数的最优偏差估计值。此外,我们提议采用全球Trefftz DG方法,结合当地DG方法,在厌食介质媒体中解决时间依赖线性线性非热波方程式。特别是,对原PDEs的(非均匀)drichlet、Neumann和混合边界条件进行误差分析。此外,还提议了一项将模型分解出不同介质介质介质介质媒介的战略。我们提议了理论结果的正确性,并表明由此得出的近似解决办法。