An Iterative Two-Grid Method of a Finite Element PML Approximation for the Two Dimensional Maxwell Problem (Advances in Applied Mathematics and Mechanics, 2018)
In this paper, we propose an iterative two-grid method for the edge finiteelement discretizations (a saddle-point system) of Perfectly Matched Layer (PML)equations to the Maxwell scattering problem in two dimensions. Firstly, we usea fine space to solve a discrete saddle-point system of $H(grad)$ variational problems,denoted by auxiliary system 1. Secondly, we use a coarse space to solve theoriginal saddle-point system. Then, we use a fine space again to solve a discrete $\boldsymbol{H}(curl)$-elliptic variational problems, denoted by auxiliary system 2. Furthermore,we develop a regularization diagonal block preconditioner for auxiliary system 1and use $H$-$X$ preconditioner for auxiliary system 2. Hence we essentially transformthe original problem in a fine space to a corresponding (but much smaller)problem on a coarse space, due to the fact that the above two preconditioners areefficient and stable. Compared with some existing iterative methods for solvingsaddle-point systems, such as PMinres, numerical experiments show the competitiveperformance of our iterative two-grid method.