Eigen-Decomposition of Discrete Curl Type Operators and Its Applications for Maxwell's Equations

In this article, we mainly consider how to design some fast algorithms for some eigenvalue problems and linear systems which are derived from Maxwell''s equations. We first started from the issue of photonic crystal, the mathematical model of photonic crystal is a special case of Maxwell equations under some simplifying assumptions,which is called time harmonic Maxwell''s equations. We refer to some photonic crystal research papers and learn some discretization methods of Maxwell''s equations. Finally, we mainly used Yee''s Scheme method, by applying Yee''s Scheme method, the time harmonic Maxwell''s equations will be transformed into a generalized eigenvalue problem, so we began to study the generalized eigenvalue problem. After some effort, we found an explicit eigen-decomposition of the double curl discrete matrix and used some matrix computation techniques to accelerate computation speed significantly, we called this algorithm null space free method. Moreover, we construct singular value decomposition of a single curl from the previous eigen-decomposition, then we used such decomposition to solve the chiral medium and plasma numerical simulation problems. Finally, we generalized the results of these decompositions, so that these techniques can be used under different boundary conditions, such as perfect matching boundary condition, Dirichlet boundary condition and quasi-periodic boundary condition, etc.