Numerische Mathematik

, Volume 58, Issue 1, pp 79–93 | Cite as

On block diagonal and Schur complement preconditioning

  • Jan Mandel


We study symmetric positive definite linear systems, with a 2-by-2 block matrix preconditioned by inverting directly one of the diagonal blocks and suitably preconditioning the other. Using an approximate version of Young's “Property A”, we show that the condition number of the Schur complement is smaller than the condition number obtained by the block-diagonal preconditioning. We also get bounds on both condition numbers from a strengthened Cauchy inequality. For systems arising from the finite element method, the bounds do not depend on the number of elements and can be obtained from element-by-element computations. The results are applied to thep-version finite element method, where the first block of variables consists of degrees of freedom of a low order.

Subject classifications

AMS(MOS) 65N20 CR: G1.8 


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Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Jan Mandel
    • 1
  1. 1.Computational Mathematics GroupUniversity of Colorado at DenverDenverUSA

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