In this paper we are concerned with the numerical method for three-dimensional anisotropic Helmholtz equations with variable wave numbers, where positive definite matrices define anisotropic media. We define novel generalized plane wave basis functions based on rigorous choice of the coordinate transformation. Then we derive the desired error estimates of the resulting approximate solutions with respect to the condition number of the coefficient matrices, under an 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 method possess high accuracies.

在本文中，我们关注具有可变波数的三维各向异性亥姆霍兹方程的数值方法，其中正定矩阵定义各向异性介质。我们基于坐标变换的严格选择定义了新颖的广义平面波基函数。然后，在假设多面体网格的形状规律性的情况下，我们根据系数矩阵的条件数推导出所得近似解的所需误差估计。数值结果验证了理论结果的有效性，表明该方法生成的近似解具有较高的精度。