The z-transform is introduced to analyze a full discretization method fora partial integro-differential equation (PIDE) with a weakly singular kernel. In thismethod, spectral collocation is used for the spatial discretization, and, for the time stepping, the finite difference method combined with the convolution quadrature rule isconsidered. The global stability and convergence properties of complete discretizationare derived and numerical experiments are reported.
In this paper,we propose an iterative two-grid method for the edge finite element discretizations(a saddle-point system)of Perfectly Matched Layer(PML)equations to the Maxwell scattering problem in two dimensions.Firstly,we use a 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 the original saddle-point system.Then,we use a fine space again to solve a discrete H(curl)-elliptic variational problems,denoted by auxiliary system 2.Furthermore,we develop a regularization diagonal block preconditioner for auxiliary system 1 and use H-X preconditioner for auxiliary system 2.Hence we essentially transform the 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 are efficient and stable.Compared with some existing iterative methods for solving saddle-point systems,such as PMinres,numerical experiments show the competitive performance of our iterative two-grid method.
Chunmei LiuShi ShuYunqing HuangLiuqiang ZhongJunxian Wang
