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A least-squares approach based on a discrete minus one inner product for first order systems


Authors: James H. Bramble, Raytcho D. Lazarov and Joseph E. Pasciak
Journal: Math. Comp. 66 (1997), 935-955
MSC (1991): Primary 65N30; Secondary 65F10
DOI: https://doi.org/10.1090/S0025-5718-97-00848-X
MathSciNet review: 1415797
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Abstract: The purpose of this paper is to develop and analyze a least-squares approximation to a first order system. The first order system represents a reformulation of a second order elliptic boundary value problem which may be indefinite and/or nonsymmetric. The approach taken here is novel in that the least-squares functional employed involves a discrete inner product which is related to the inner product in $H^{-1}(\Omega )$ (the Sobolev space of order minus one on $\Omega $). The use of this inner product results in a method of approximation which is optimal with respect to the required regularity as well as the order of approximation even when applied to problems with low regularity solutions. In addition, the discrete system of equations which needs to be solved in order to compute the resulting approximation is easily preconditioned, thus providing an efficient method for solving the algebraic equations. The preconditioner for this discrete system only requires the construction of preconditioners for standard second order problems, a task which is well understood.


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Additional Information

James H. Bramble
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853 and Department of Mathematics, Texas A&M University, College Station, Texas 77843-3404
Email: bramble@math.tamu.edu

Raytcho D. Lazarov
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3404
Email: lazarov@math.tamu.edu

Joseph E. Pasciak
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3404
Email: pasciak@math.tamu.edu

DOI: https://doi.org/10.1090/S0025-5718-97-00848-X
Received by editor(s): October 9, 1995
Received by editor(s) in revised form: June 5, 1996
Additional Notes: This manuscript has been authored under contract number DE-AC02-76CH00016 with the U.S. Department of Energy. Accordingly, the U.S. Government retains a non-exclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes. This work was also supported in part under the National Science Foundation Grant No. DMS-9007185 and by the U.S. Army Research Office through the Mathematical Sciences Institute, Cornell University.
Article copyright: © Copyright 1997 American Mathematical Society

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