Deflation and projection methods applied to symmetric positive semi-definite systems

Journal article (2016)

Authors

Elisabeth Ludwig Oberstufen-Kolleg Bielefeld
R Nabben Technical University of Berlin
J.M. Tang Numerical Analysis - EEMCS
Research Group
Numerical Analysis (EEMCS) (TU Delft)
Multigrid Deflation Preconditioning Domain decomposition Conjugate gradients Two-grid schemes Projection methods Coarse-grid corrections SPSD matrices
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Published Date

15-01-2016

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Numerical Analysis

Abstract

Linear systems with a singular symmetric positive semi-definite matrix appear frequently in practice. This usually does not lead to difficulties for CG methods as long as these systems are consistent. However, the construction of a preconditioner, especially the construction of two-level and multilevel methods, becomes more complicated, since singular coarse grid matrices or Galerkin matrices may occur. Here we continue the work started in [21,22] where deflation is used for some special singular coefficient matrices. Here we show that deflation and other projection-type preconditioners can be applied to arbitrary singular problems without any difficulties. In each of these methods, a two-level preconditioner is involved where coarse-grid systems based on a singular Galerkin matrix should be solved. We prove that each projection operator consisting of a singular Galerkin matrix can be written as an operator with a nonsingular Galerkin matrix. Therefore many results that hold for nonsingular Galerkin matrices are also valid for problems with singular Galerkin matrices.