Comparison of Two-Level Preconditioners Derived from Deflation, Domain Decomposition and Multigrid Methods

For various applications, it is well-known that a multi-level, in particular twolevel, preconditioned CG (PCG) method is an efficient method for solving large and sparse linear systems with a coefficient matrix that is symmetric positive definite. The corresponding two-level preconditioner combines traditional and projection-type preconditioners to get rid of the effect of both small and large eigenvalues of the coefficient matrix. In the literature, various two-level PCG methods are known, coming from the fields of deflation, domain decomposition and multigrid. Even though these two-level methods differ a lot in their specific components, it can be shown that from an abstract point of view they are closely related to each other. We investigate their equivalences, robustness, spectral and convergence properties, by accounting for their implementation, the effect of roundoff errors and their sensitivity to inexact coarse solves, severe termination criteria and perturbed starting vectors. Part of this work has been done during the visit of the first, third and fourth author at Technische Universität Berlin. The research is partially funded by the Dutch BSIK/BRICKS project and the Deutsche Forschungsgemeinschaft (DFG), Project NA248/2-2. J.M. Tang · C. Vuik ( ) Faculty of Electrical Engineering, Mathematics and Computer Science, Delft Institute of Applied Mathematics, Delft University of Technology, J.M. Burgerscentrum, Mekelweg 4, 2628 CD Delft, The Netherlands e-mail: [email protected] J.M. Tang e-mail: [email protected] R. Nabben Institut für Mathematik, Technische Universität Berlin, MA 3-3, Straße des 17. Juni 136, 10623 Berlin, Germany e-mail: [email protected] Y.A. Erlangga Department of Earth and Ocean Sciences, The University of British Columbia, 6339 Stores Road, Vancouver, British Columbia, V6T 1Z4, Canada e-mail: [email protected] J Sci Comput (2009) 39: 340–370 341