A scalable Helmholtz solver combining the shifted Laplace preconditioner with multigrid deflation

A scalable Helmholtz solver combining the shifted Laplace preconditioner with multigrid deflation

Sheikh, A.H.
Lahaye, D.
Vuik, C.

Electrical Engineering, Mathematics and Computer Science

2011-01-31

A Helmholtz solver whose convergence is parameter independent can be obtained by combining the shifted Laplace preconditioner with multigrid deflation. To proof this claim, we develop a Fourier analysis of a two-level variant of the algorithm proposed in [1]. In this algorithm those eigenvalues that prevent the shifted Laplace preconditioner from being scalable are removed by deflation using multigrid vectors. Our analysis shows that the spectrum of the two-grid operator consists of a cluster surrounded by a few outliers, yielding a number of outer Krylov subspace iterations that remains constant as the wave number increases. Our analysis furthermore shows that the imaginary part of the shift in the two-grid operator can be made arbitrarily large without affecting the convergence. This opens promising perspectives on obtaining a very good preconditioner at very low cost. Numerical tests for problems with constant and non-constant wave number illustrate our convergence theory.

http://resolver.tudelft.nl/uuid:5cd1678a-4ac8-4490-aa3c-c91584c16be9

Delft University of Technology, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft Institute of Applied Mathematics

1389-6520

Reports of the Department of Applied Mathematical Analysis, 11-01

Institutional Repository

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(c)2011 Sheikh, A.H., Lahaye, D., Vuik, C.