Numerische Mathematik

, Volume 62, Issue 1, pp 413–438

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

  • Sze-Ping Wong

DOI: 10.1007/BF01396237

Cite this article as:
Wong, SP. Numer. Math. (1992) 62: 413. doi:10.1007/BF01396237


This work deals with theH1 condition numbers and the distribution of theBh singular values of the preconditioned operators {Bh−1Ah}0<h<1, whereAh andBh are finite element discretizations of second order elliptic operators,A andB respectively.B is also assumed to be self-adjoint and positive definite. For conforming finite elements, Parter and Wong have shown that the singular values “cluster” in a positive finite interval. Goldstein also has derived results on the spectral distribution ofBh−1Ah using a different approach. As a generalization of the results of Parter and Wong, the current work includes nonconforming finite element methods which deal with Dirichlet boundary conditions. It will be shown that, in this more general setting, the singular values also “cluster” in a positive finite 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. Finally, it will be shown that the same results can be proven by an approach generalized from the work of Goldstein.

Mathematics Subject Classification (1991)

65N22 65N30 

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

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

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