Summary
This work deals with theL 2 condition numbers and the distribution of theL 2 singular values of the preconditioned operators {B −1h 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 −1 h A h “cluster” and “fill-in” the interval [θ min,θ max], where 0<θ min≦θ max 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, theL 2 condition numbers of {B −1 h 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.
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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. I. Numer. Math. 62, 391–411 (1992). https://doi.org/10.1007/BF01396236
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DOI: https://doi.org/10.1007/BF01396236