A shifted Laplacian operator is obtained from the Helmholtz operator by adding a complex damping.It serves as a basic tool in the most successful multigrid approach for solving highly indefinite Helmholtz equations—a...A shifted Laplacian operator is obtained from the Helmholtz operator by adding a complex damping.It serves as a basic tool in the most successful multigrid approach for solving highly indefinite Helmholtz equations—a Shifted Laplacian preconditioner for Krylov-type methods.Such preconditioning significantly acceler-ates Krylov iterations,much more so than the multigrid based on original Helmholtz equations.In this paper,we compare approximation and relaxation properties of the Helmholtz operator with and without the complex shift,and,based on our ob-servations,propose a new hybrid approach that combines the two.Our analytical conclusions are supported by two-dimensional numerical results.展开更多
文摘A shifted Laplacian operator is obtained from the Helmholtz operator by adding a complex damping.It serves as a basic tool in the most successful multigrid approach for solving highly indefinite Helmholtz equations—a Shifted Laplacian preconditioner for Krylov-type methods.Such preconditioning significantly acceler-ates Krylov iterations,much more so than the multigrid based on original Helmholtz equations.In this paper,we compare approximation and relaxation properties of the Helmholtz operator with and without the complex shift,and,based on our ob-servations,propose a new hybrid approach that combines the two.Our analytical conclusions are supported by two-dimensional numerical results.
