This paper concerns the preconditioning technique for discrete systems arising from time-harmonic Maxwell equations with absorptions, where the discrete systems are generated by N\'ed\'elec finite element methods of fixed order on meshes with suitable size. This kind of preconditioner is defined as a two-level hybrid form, which falls into the class of ``unsymmetrically weighted'' Schwarz method based on the overlapping domain decomposition with impedance boundary subproblems. The coarse space in this preconditioner is constructed by specific eigenfunctions solving a series of generalized eigenvalue problems in the local discrete Maxwell-harmonic spaces according to a user-defined tolerance $\rho$. We establish a stability result for the considered discrete variational problem. Using this discrete stability, we prove that the two-level hybrid Schwarz preconditioner is robust in the sense that the convergence rate of GMRES is independent of the mesh size, the subdomain size and the wave-number when $\rho$ is chosen appropriately. We also define an economical variant that avoids solving generalized eigenvalue problems. Numerical experiments confirm the theoretical results and illustrate the efficiency of the preconditioners.

翻译：本文研究带吸收项的时谐Maxwell方程离散系统的预处理技术，其中离散系统由固定阶数的Nédélec有限元方法在适当尺寸的网格上生成。此类预处理器定义为双层混合形式，属于基于阻抗边界子问题的重叠区域分解的"非对称加权"Schwarz方法范畴。该预处理器中的粗空间由特定特征函数构建，这些特征函数根据用户定义的容差ρ在局部离散Maxwell-调和空间中求解一系列广义特征值问题。我们为所考虑的离散变分问题建立了稳定性结果。利用该离散稳定性，我们证明当ρ选择适当时，双层混合Schwarz预处理器的收敛性具有鲁棒性，即GMRES的收敛速率与网格尺寸、子域尺寸及波数无关。我们还定义了一种避免求解广义特征值问题的经济型变体。数值实验验证了理论结果，并说明了预处理器的有效性。