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Numerische Mathematik

, Volume 62, Issue 1, pp 391–411 | Cite as

Preconditioning nonconforming finite element methods for treating Dirichlet boundary conditions. I

  • Sze-Ping Wong
Article

Summary

This work deals with theL2 condition numbers and the distribution of theL2 singular values of the preconditioned operators {B h −1 Ah}0<h<1, whereA h andB h are finite element discretizations of second order elliptic operators,A andB respectively. For conforming finite elements, it was shown in the work of Goldstein, Manteuffel and Parter that if the leading part ofB is a scalar multiple (1/Θ) of the leading part ofA, then the singular values ofB h −1 A h “cluster” and “fill-in” the interval [θmin,θmax], where 0<θminθmax are the minimum and maximum of the factor Θ. As a generalization of these results, the current work includes nonconforming finite element methods which deal with Dirichlet boundary conditions. It will be shown that, in this more general setting, theL2 condition numbers of {B h −1 A h } are uniformly bounded. Moreover, the singular values also “cluster” and “fill-in” the same interval. In particular, if the leading part ofB is the same as the leading part ofA, then the singular values cluster about the point {1}. Two specific methods are given as applications of this theory. They are the penalty method of Babuška and the method of “nearly zero” boundary conditions of Nitsche.

Mathematics Subject Classification (1991)

65N22 65N30 

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References

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Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Sze-Ping Wong
    • 1
  1. 1.Department of MathematicsUniversity of Wisconsin-MadisonMadisonUSA

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