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1 results

  • Use of Shifted Laplacian Operators for Solving Indefinite Helmholtz Equations

  • Ira Livshits
  • Journal:
    Numerical Mathematics: Theory, Methods and Applications / Volume 8 / Issue 1 / February 2015
    Published online by Cambridge University Press:
    03 March 2015, pp. 136-148
    Print publication:
    February 2015

  • 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.