GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Matrix-dependent prolongations and restrictions in a blackbox multigrid solver
Journal of Computational and Applied Mathematics
SIAM Journal on Scientific and Statistical Computing
Dispersion-relation-preserving finite difference schemes for computational acoustics
Journal of Computational Physics
Journal of Computational Physics
Flexible Multiple Semicoarsening for Three-Dimensional Singularly Perturbed Problems
SIAM Journal on Scientific Computing
Preconditioning techniques for large linear systems: a survey
Journal of Computational Physics
SIAM Journal on Numerical Analysis
On a class of preconditioners for solving the Helmholtz equation
Applied Numerical Mathematics
A perfectly matched layer for the Helmholtz equation in a semi-infinite strip
Journal of Computational Physics
A Novel Multigrid Based Preconditioner For Heterogeneous Helmholtz Problems
SIAM Journal on Scientific Computing
Compact finite difference schemes of sixth order for the Helmholtz equation
Journal of Computational and Applied Mathematics
A parallel multigrid-based preconditioner for the 3D heterogeneous high-frequency Helmholtz equation
Journal of Computational Physics
Spectral Analysis of the Discrete Helmholtz Operator Preconditioned with a Shifted Laplacian
SIAM Journal on Scientific Computing
Journal of Computational Physics
Discontinuous Galerkin Methods for the Helmholtz Equation with Large Wave Number
SIAM Journal on Numerical Analysis
A perfectly matched layer for the absorption of electromagnetic waves
Journal of Computational Physics
An optimal 25-point finite difference scheme for the Helmholtz equation with PML
Journal of Computational and Applied Mathematics
Hi-index | 31.45 |
In this paper, a new 27-point finite difference method is presented for solving the 3D Helmholtz equation with perfectly matched layer (PML), which is a second order scheme and pointwise consistent with the equation. An error analysis is made between the numerical wavenumber and the exact wavenumber, and a refined choice strategy based on minimizing the numerical dispersion is proposed for choosing weight parameters. A full-coarsening multigrid-based preconditioned Bi-CGSTAB method is developed for solving the linear system stemming from the Helmholtz equation with PML by the finite difference scheme. The shifted-Laplacian is extended to precondition the 3D Helmholtz equation, and a spectral analysis is given. The discrete preconditioned system is solved by the Bi-CGSTAB method, with a multigrid method used to invert the preconditioner approximately. Full-coarsening multigrid is employed, and a new matrix-based prolongation operator is constructed accordingly. Numerical results are presented to demonstrate the efficiency of both the new 27-point finite difference scheme with refined parameters, and the preconditioned Bi-CGSTAB method with the 3D full-coarsening multigrid.