In this paper we are concerned with plane wave discontinuous Galerkin (PWDG) methods for Helmholtz equation and time-harmonic Maxwell equations in three-dimensional anisotropic media, for which the coefficients of the equations are matrices instead of numbers. We first define novel plane wave basis functions based on rigorous choices of scaling transformations and coordinate transformations. Then we derive the error estimates of the resulting approximate solutions with respect to the condition number of the coefficient matrices, under a new assumption on the shape regularity of polyhedral meshes. Numerical results verify the validity of the theoretical results, and indicate that the approximate solutions generated by the proposed PWDG method possess high accuracy.

在本文中，我们讨论了三维各向异性介质中 Helmholtz 方程和时谐麦克斯韦方程的平面波不连续伽辽金 (PWDG) 方法，其中方程的系数是矩阵而不是数字。我们首先根据缩放变换和坐标变换的严格选择来定义新的平面波基函数。然后，在对多面体网格形状规律性的新假设下，我们根据系数矩阵的条件数推导出所得近似解的误差估计。数值结果验证了理论结果的有效性，表明所提出的 PWDG 方法生成的近似解具有较高的精度。