A multigrid-based preconditioned solver for the Helmholtz equation with a discretization by 25-point difference scheme

作者：Cheng, Dongsheng; Liu, Zhiyong^*; Wu, Tingting

来源：Mathematics and Computers in Simulation, 2015, 117: 54-67.

DOI：10.1016/j.matcom.2015.01.009

• Helmholtz equation
• Preconditioner
• Multigrid
• Prolongation operator

In this paper, a preconditioned iterative method is developed to solve the Helmholtz equation with perfectly matched layer (Helmholtz-PML equation). The complex shifted-Laplacian is generalized to precondition the Helmholtz-PML equation, which is discretized by an optimal 25-point finite difference scheme that we presented in Chen et al. (2011). A spectral analysis is given for the discrete preconditioned system from the perspective of linear fractal mapping, and Bi-CGSTAB is used to solve it. The multigrid method is employed to invert the preconditioner approximately, and a new matrix-based prolongation operator is constructed in the multigrid cycle. Numerical experiments are presented to illustrate the efficiency of the multigrid-based preconditioned Bi-CGSTAB method with the new prolongation operator. Numerical results are also given to compare the performance of the new prolongation operator with that of the prolongation operator based on the algebraic multigrid (AMG) principle.

• 出版日期2015-11
• 单位山东师范大学; 深圳职业技术学院; 深圳信息职业技术学院