Bollhöfer, Matthias; Grote, Marcus J.; Schenk, Olaf Algebraic multilevel preconditioner for the Helmholtz equation in heterogeneous media. (English) Zbl 1203.65273 SIAM J. Sci. Comput. 31, No. 5, 3781-3805 (2009). The 2-level multigrid method and the 2-level deflation method have a common property. It is assumed that the restriction to a certain subspace admits an easy solution. In the framework of multigrid, the Richardson iteration provides a good smoother, and one may understand that the condition number is reduced for the complementary space. This reduction of the condition number is used in the framework of deflation for a preconditioning. After the algebraic relations for the comparison of the two methods are provided, many numerical examples are discussed. Reviewer: Dietrich Braess (Bochum) Cited in 33 Documents MSC: 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs 65F10 Iterative numerical methods for linear systems 78A45 Diffraction, scattering 86A15 Seismology (including tsunami modeling), earthquakes 65F08 Preconditioners for iterative methods 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 65N22 Numerical solution of discretized equations for boundary value problems involving PDEs Keywords:algebraic multigrid; deflation; graph-pivoting; Helmholtz equation; inhomogeneous media; symmetric indefinite matrix; inverse-based pivoting; Richardson iteration; condition number; preconditioning; numerical examples Software:SuperLU-DIST; QMRPACK PDFBibTeX XMLCite \textit{M. Bollhöfer} et al., SIAM J. Sci. Comput. 31, No. 5, 3781--3805 (2009; Zbl 1203.65273) Full Text: DOI