Fast Maxwell Solvers Based on Exact Discrete Eigen-Decompositions I. Two-Dimensional Case

In this paper, we propose fast solvers for Maxwell's equations in rectangular domains. We first discretize the simplified Maxwell's eigenvalue problems by employing the lowest-order rectangular N\'ed\'elec elements and derive the discrete eigen-solutions explicitly, providing a Hodge-Helmholtz decomposition framework at the discrete level. Based on exact eigen-decompositions, we further design fast solvers for various Maxwell's source problems, guaranteeing either the divergence-free constraint or the Gauss's law at the discrete level. With the help of fast sine/cosine transforms, the computational time grows asymptotically as $\mathcal{O}(n^2\log n)$ with $n$ being the number of grids in each direction. Our fast Maxwell solvers outperform other existing Maxwell solvers in the literature and fully rival fast scalar Poisson/Helmholtz solvers based on trigonometric transforms in either efficiency, robustness, or storage complexity. It is also utilized to perform an efficient pre-conditioning for solving Maxwell's source problems with variable coefficients. Finally, numerical experiments are carried out to illustrate the effectiveness and efficiency of the proposed fast solver.