Title: The PML Method and Fast Solvers for the Helmholtz Equation in Unbounded Domain
Speaker: Pro.Chen Zhiming
Head of Institute of Computational Mathematics and Scientific/Engineering Computing
Abstract: In this talk we first introduce the adaptive perfectly matched layer (PML) method for solving Helmholtz scattering problems. The adaptive PML method provides a complete numerical strategy to solve the scattering problems in the framework of finite element which produces automatically a coarse mesh size away from the fixed domain and thus makes the total computational costs insensitive to the thickness the absorbing PML layers. Next we propose a domain decomposition method for solving the truncated PML approximation in bounded domain of Helmholtz scattering problems. The method is based on the decomposition of the domain into non-overlapping layers and the idea of source transfer which transfers the sources equivalently layer by layer so that the solution in the final layer can be solved using a PML method defined locally outside the last two layers. The convergence of the method is proved for the case of constant wave number based on the analysis of fundamental solutions of the PML equation. The method can be used as an efficient preconditioner in the preconditioned GMRES method for solving discrete Helmholtz equations with constant and heterogeneous wave numbers. Numerical examples are included.