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Extra resources for Advanced Computational Methods in Science and Engineering
This is due to the imaginary term, β2 i in the shifted Laplacian. LFA also indicates that Mh based on the fourth-order discretization can be dealt with in multigrid as efficiently as the second-order discretization. 4, and obtain an efficient preconditioned Krylov subspace method. 5). 5 was chosen for the high wavenumber problems. The other multigrid components were: • Restriction operators, IhH , based on 2D full weighting, whose stencil  reads: H 121 ∧ 1 2 4 2 , IhH = 16 121 h (11) with h denoting the fine-, and H denoting the coarse-mesh size.
E. Plessix van Gijzen for their input in the project. P. MacLachlan and N. Umetani for their contribution to this article, as the present article relies on paper  of which they are the co-authors. N. Umetani performed some additional numerical experiments, especially for this paper. References 1. T. A IRAKSINEN , E. H EIKKOLA , A. P ENNANEN , J. T OIVANEN , An algebraic multigrid based shifted-Laplacian preconditioner for the Helmholtz equation. J. Comp. Physics, 226: 1196–1210, 2007. 2. R.
S UH , An optimal 9-point, finite-difference, frequency space, 2-D scalar wave extrapolator, Geophysics 61(2): 529–537, 1996. 30. R. K ETTLER , Analysis and comparison of relaxation schemes in robust multigrid and preconditioned conjugate gradient methods. In:W. Hackbusch, U. ), Multigrid methods, Lecture Notes in Mathematics 960: 502–534, Springer, Berlin, 1982. 31. M. K HALIL , Analysis of linear multigrid methods for elliptic differential equations with discontinuous and anisotropic coefficients.