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

Abstract

For various applications, it is well-known that a multi-level, in particular two-level, 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.

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Correspondence to C. Vuik.

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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.

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Tang, J.M., Nabben, R., Vuik, C. et al. Comparison of Two-Level Preconditioners Derived from Deflation, Domain Decomposition and Multigrid Methods. J Sci Comput 39, 340–370 (2009). https://doi.org/10.1007/s10915-009-9272-6

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  • DOI: https://doi.org/10.1007/s10915-009-9272-6

Keywords

  • Deflation
  • Domain decomposition
  • Multigrid
  • Conjugate gradients
  • Two-grid schemes
  • Two-level preconditioning
  • SPD matrices
  • Two-level PCG methods