http://math.ucdenver.edu/~aknyazev
Department of Mathematics, University of Colorado Denver
P.O. Box 173364, Campus Box 170, Denver, CO 80217-3364.
Street Address: 1250 14th Str. Room 644, Denver CO 80202
Phone: (303) 556-8442. Fax: (303) 556-8550
Email: aknyazev@math.ucdenver.edu
 
Andrew V. Knyazev (speaker) and Olof B. Widlund
 
Uniform Finite Element Error Estimates for
Differential Equations with Rough Coefficients

We consider a parametric family of boundary value problems for the diffusion equation with the 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 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.

The talk is based on the following paper: A. V. Knyazev and Olof Widlund, Lavrentiev Regularization + Ritz Approximation = Uniform Finite Element Error Estimates for Differential Equations with Rough Coefficients. Submitted to Mathematics of Computation. Published as a technical report UCD-CCM 132, 1998, at the Center for Computational Mathematics, University of Colorado Denver

 
 

STATE OF THE ART IN FINITE ELEMENT METHOD
     July 21 --24 1998 City University of Hong Kong, Hong Kong