SIAM Journal on Mathematical Analysis, Volume 53, Issue 3, Page 2691-2717, January 2021.
We consider the time-harmonic Maxwell's equations with physical parameters, namely, the electric permittivity and the magnetic permeability, that are complex, possibly non-Hermitian, tensor fields. Both tensor fields verify a general ellipticity condition. In this work, the well-posedness of formulations for the Dirichlet and Neumann problems (i.e., with a boundary condition on the electric field or its curl, respectively) is proven using well-suited function spaces and Helmholtz decompositions. For both problems, the a priori regularity of the solution and the solution's curl is analyzed. The regularity results are obtained by splitting the fields and using shift theorems for second-order divergence elliptic operators. Finally, the discretization of the formulations with a $H$(curl)-conforming approximation based on edge finite elements is considered. An a priori error estimate is derived and verified thanks to numerical results with an elementary benchmark.

中文翻译：

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各向异性介质中时谐波电磁问题的变式和低规则解分析

SIAM数学分析杂志，第53卷，第3期，第2691-2717页，2021年1月。
我们考虑具有物理参数（即介电常数和磁导率）的时谐麦克斯韦方程组，它们是复杂的，可能是非Hermitian的张量场。两个张量场都验证了一般的椭圆率条件。在这项工作中，使用合适的函数空间和亥姆霍兹分解证明了Dirichlet和Neumann问题的公式的适定性（即分别具有电场或其卷曲的边界条件）。对于这两个问题，分析了溶液的先验规律性和溶液的卷曲度。通过拆分字段并使用二阶散度椭圆算子的移位定理获得正则性结果。最后，考虑采用基于边缘有限元的符合$ H $（curl）的近似式离散化。先验误差估计的获得和验证归功于具有基本基准的数值结果。