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Volume 8, Issue 1
Use of Shifted Laplacian Operators for Solving Indefinite Helmholtz Equations

Ira Livshits

Numer. Math. Theor. Meth. Appl., 8 (2015), pp. 136-148.

Published online: 2015-08

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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.

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@Article{NMTMA-8-136, author = {}, title = {Use of Shifted Laplacian Operators for Solving Indefinite Helmholtz Equations}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2015}, volume = {8}, number = {1}, pages = {136--148}, 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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2015.w03si}, url = {http://global-sci.org/intro/article_detail/nmtma/12403.html} }
TY - JOUR T1 - Use of Shifted Laplacian Operators for Solving Indefinite Helmholtz Equations JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 136 EP - 148 PY - 2015 DA - 2015/08 SN - 8 DO - http://doi.org/10.4208/nmtma.2015.w03si UR - https://global-sci.org/intro/article_detail/nmtma/12403.html KW - AB -

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.

Ira Livshits. (2020). Use of Shifted Laplacian Operators for Solving Indefinite Helmholtz Equations. Numerical Mathematics: Theory, Methods and Applications. 8 (1). 136-148. doi:10.4208/nmtma.2015.w03si
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