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Lavrentiev Regularization + Ritz Approximation Uniform Finite Element Error
Estimates for Differential Equations with Rough Coefficients

Andrew Knyazev and Olof Widlund

May 1, 1998

URN = ncstrl.cudenver_ccm/UCD-CCM-132

URL =
http://cs-tr.cs.cornell.edu:80/Dienst/UI/1.0/Display/ncstrl.cudenver_ccm/UCD-CCM-132

**Abstract: **
We consider a parametric family of boundary value problems for the diffusion
equation with the diffusion coefficient eual 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 consider 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. 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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