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Extra info for Advanced Computational Methods in Science and Engineering (Lecture Notes in Computational Science and Engineering)
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. D. Thesis, Delft University of Technology, Delft, Netherlands, 1989. 32. S. K IM , S. K IM , Multigrid simulation for high-frequency solutions of the Helmholtz problem in heterogeneous media, SIAM J. Sci. Comput. 24: 684–701, 2002. 33. D. L AHAYE , H. D E G ERSEM , S.
Bi-CGSTAB is based on the idea of computing two mutually bi-orthogonal bases for the Krylov subspaces based on matrix, Ah , and its conjugate transpose, AH h and is easy to implement. Without a preconditioner, however, the Krylov subspace methods converge very slowly, or not at all, for the problems of interest . By preconditioning with a matrix, Mh−1 , we solve an equivalent linear system, Ah Mh−1 φ˜h = bh , φ˜h = Mh φh . (8) Shifted-Laplacian Preconditioners for Heterogeneous Helmholtz Problems 31 The challenge, then, is to find a matrix, Mh , such that Ah Mh−1 has a spectrum that is favorable for iterative solution with Krylov subspace methods, and whose inverse, Mh−1 , can be efficiently approximated.
125, the second- and fourth-order discretizations. Coarse grids consisting of 322 , 482 and 642 points are chosen. As shown, the profile lines generated by the fourth-order discretization converge very nicely towards the physical solution (7), whereas the profile with the second-order discretization on the 322 -grid is too inaccurate to show, and the solutions on the 482 − and 642 −grids are shifted in phase. 625 is not satisfied there. 3 Iterative solution method Before we discuss the solution method of choice, we outline the general convergence problems when using the multigrid method directly for the discrete Helmholtz equation of interest.