Summary
This work deals with theH 1 condition numbers and the distribution of theB h singular values of the preconditioned operators {B −1 h A h }0<h<1, whereA h andB h 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 ofB −1 h A h 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.
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This research was supported by the National Science Foundation under grant number DMS-8913091.
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Wong, SP. Preconditioning nonconforming finite element methods for treating Dirichlet boundary conditions. II. Numer. Math. 62, 413–438 (1992). https://doi.org/10.1007/BF01396237
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DOI: https://doi.org/10.1007/BF01396237