Use of Shifted Laplacian Operators for Solving Indefinite Helmholtz Equations. Issue 1 (3rd March 2015)

• Record Type: Journal Article
• Authors: Livshits, Ira
• Is Part Of: Numerical mathematics. Volume 8:Issue 1(2015)
• Page Start: 136
• Page End: 148
• Publication Date: 2015-03-03
• Subjects: 65F10, -- 65N22, -- 65N55; Indefinite Helmholtz operator, -- multigrid, -- shifted Laplacian, -- ray correction; Numerical analysis -- Periodicals; Numerical analysis; Periodicals; 518.05
• DOI: 10.4208/nmtma.2015.w03si
• Languages: English
• ISSNs: 1004-8979

Abstract

Abstract: 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 accelerates 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 observations, propose a new hybrid approach that combines the two. Our analytical conclusions are supported by two-dimensional numerical results.