## 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.

Original language | English |
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Pages (from-to) | 253-273 |

Number of pages | 21 |

Journal | Linear Algebra and Its Applications |

Volume | 489 |

DOIs | |

Publication status | Published - 15 Jan 2016 |

## Keywords

- Coarse-grid corrections
- Conjugate gradients
- Deflation
- Domain decomposition
- Multigrid
- Preconditioning
- Projection methods
- SPSD matrices
- Two-grid schemes