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On the convergence of shifted Laplace preconditioner combined with multilevel deflation


Correspondence to: A. H. Sheikh, Delft Institute of Applied Mathematics, Delft University of Technology, Mekelweg 4, Delft, Netherlands 2628CD.



Deflating the shifted Laplacian with geometric multigrid vectors yields speedup. To verify this claim, we investigate a simplified variant of Erlangga and Nabben presented in [Erlangga and Nabben, ETNA, 2008;31:403–424]. We derive expressions for the eigenvalues of the two-level preconditioner for the one-dimensional problem. These expressions show that the algorithm analyzed is not scalable. They also show that the imaginary shift can be increased without delaying the convergence of the outer Krylov acceleration. An increase of the number of grid points per wavelength results in convergence acceleration. This contrasts to the use of the shifted Laplace preconditioner. Our analysis also shows that the use of deflation results in a spectrum more favorable to the convergence of the outer Krylov acceleration. The near-null space components are still insufficiently well resolved, and the number of iterations increases with the wavenumber. In the two-dimensional case, the number of near-zero eigenvalues is larger than in the one-dimensional case. We perform numerical computations with the two-level and multilevel versions of the algorithm on constant and nonconstant wavenumber problems. Our numerical results confirm our spectral analysis. Copyright © 2013 John Wiley & Sons, Ltd.