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

DOI:
https://doi.org/10.1090/S0025-5718-01-01378-3

Published electronically:
July 13, 2001

MathSciNet review:
1933812

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Abstract | References | Similar Articles | Additional Information

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:
https://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