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

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## Summary

This work deals with the*L*^{2} condition numbers and the distribution of the*L*^{2} singular values of the preconditioned operators {*B* _{h} ^{−1} A_{h}}_{0<h<1}, where*A*_{ h } and*B*_{ h } are finite element discretizations of second order elliptic operators,*A* and*B* respectively. For conforming finite elements, it was shown in the work of Goldstein, Manteuffel and Parter that if the leading part of*B* is a scalar multiple (1/Θ) of the leading part of*A*, then the singular values of*B* _{ h } ^{−1} *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, the*L*^{2} 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 of*B* is the same as the leading part of*A*, 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## Preview

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## References

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