ISSN:
0945-3245
Keywords:
65N22
;
65N30
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Summary This work deals with theL 2 condition numbers and the distribution of theL 2 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 [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 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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01396236
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