The time-harmonic Maxwell equations are used to study the effect of electric and magnetic fields on each other. Although the linear systems resulting from solving this system using FEMs are sparse, direct solvers cannot reach the linear complexity. In fact, due to the indefinite system matrix, iterative solvers suffer from slow convergence. In this work, we study the effect of using the inverse of $\CH$-matrix approximations of the Galerkin matrices arising from N\'ed\'elec's edge FEM discretization to solve the linear system directly. We also investigate the impact of applying an $\mathcal{H}-LU$ factorization as a preconditioner and we study the number of iterations to solve the linear system using iterative solvers.

翻译：时谐麦克斯韦方程组用于研究电场与磁场的相互作用。尽管采用有限元方法求解该方程组产生的线性系统具有稀疏性，但直接求解器无法达到线性复杂度。实际上，由于系统矩阵的不定性，迭代求解器收敛缓慢。本文研究了利用Nédélec边有限元离散所得Galerkin矩阵的$\mathcal{H}$-矩阵逼近逆直接求解线性系统的效果，同时探讨了采用$\mathcal{H}-LU$分解作为预处理器的性能影响，并分析了迭代求解器处理该线性系统所需的迭代步数。