Abstract
This work deals with theH 1 condition numbers and the distribution of theB-singular values of the preconditioned operators {B −1h Ah}0{ith<1, whereA h andB h are conforming finite element discretizations of second-order elliptic operatorsA andB, respectively.B is also assumed to be self-adjoint and positive definite. As an improvement of the results of Manteuffel and Parter (1990), we give here an interval on which these singular valuescluster. Such an interval is called aneffective asymptotic range in this paper. In particular, if the leading part ofB is the same as the leading part ofA, then the singular values cluster aboutσ=“1.” Finally, we discuss the effects of replacingB −1 h by an approximation.
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Dedicated to Saul Abarbandel on the occasion of this 60th birthday.
This research was supported by the National Science Foundation under grant No. DMS-8913091.
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Parter, S.V., Wong, SP. Preconditioning second-order elliptic operators: Condition numbers and the distribution of the singular values. J Sci Comput 6, 129–157 (1991). https://doi.org/10.1007/BF01062117
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DOI: https://doi.org/10.1007/BF01062117