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Lavrentiev regularization + Ritz approximation = uniform finite element error estimates for differential equations with rough coefficients


Authors: Andrew Knyazev and Olof Widlund
Journal: Math. Comp. 72 (2003), 17-40
MSC (2000): Primary 65N30, 35R05; Secondary 35J25, 35J70
Published electronically: July 13, 2001
MathSciNet review: 1933812
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Abstract:

We consider a parametric family of boundary value problems for a diffusion equation with a diffusion coefficient equal to a small constant in a subdomain. Such problems are not uniformly well-posed when the constant gets small. However, in a series of papers, Bakhvalov and Knyazev have suggested a natural splitting of the problem into two well-posed problems. Using this idea, we prove a uniform finite element error estimate for our model problem in the standard parameter-independent Sobolev norm. We also study uniform regularity of the transmission problem, needed for approximation. A traditional finite element method with only one additional assumption, namely, that the boundary of the subdomain with the small coefficient does not cut any finite element, is considered.

One interpretation of our main theorem is in terms of regularization. Our FEM problem can be viewed as resulting from a Lavrentiev regularization and a Ritz-Galerkin approximation of a symmetric ill-posed problem. Our error estimate can then be used to find an optimal regularization parameter together with the optimal dimension of the approximation subspace.


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Additional Information

Andrew Knyazev
Affiliation: Department of Mathematics, University of Colorado at Denver P.O. Box 173364, Campus Box 170, Denver, Colorado 80217-3364
Email: knyazev@na-net.ornl.gov

Olof Widlund
Affiliation: Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, New York 10012
Email: widlund@cs.nyu.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-01-01378-3
Keywords: Galerkin, Lavrentiev, Ritz, Tikhonov, discontinuous coefficients, error estimate, finite elements, regularization, regularity, transmission problem, fictitious domain, embedding
Received by editor(s): May 19, 1998
Received by editor(s) in revised form: December 28, 2000
Published electronically: July 13, 2001
Additional Notes: The first author was supported by NSF Grant DMS-9501507
The second author was supported in part by NSF Grant CCR-9732208 and in part by the U.S. Department of Energy under contract DE-FG02-92ER25127
Article copyright: © Copyright 2001 American Mathematical Society