In this paper, we first consider the time-harmonic Maxwell equations with Dirichlet boundary conditions in three-dimensional anisotropic media, where the coefficients of the equations are general symmetric positive definite matrices. By using scaling transformations and coordinate transformations, we build the desired stability estimates between the original electric field and the transformed nonphysical field on the condition number of the anisotropic coefficient matrix. More importantly, we prove that the resulting approximate solutions generated by plane wave least squares (PWLS) methods have the nearly optimal L² error estimates with respect to the condition number of the coefficient matrix. Finally, numerical results verify the validity of the theoretical results, and the comparisons between the proposed PWLS method and the existing PWDG method are also provided.

在本文中，我们首先考虑在三维各向异性介质中具有Dirichlet边界条件的时谐Maxwell方程，其中方程的系数为一般对称正定矩阵。通过使用缩放变换和坐标变换，我们在各向异性系数矩阵的条件数上，在原始电场和变换后的非物理场之间建立了所需的稳定性估计。更重要的是，我们证明了通过平面波最小二乘（PWLS）方法生成的近似解具有接近最佳的L ²关于系数矩阵的条件数的误差估计。最后，数值结果验证了理论结果的正确性，并与提出的PWLS方法和现有的PWDG方法进行了比较。