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 regularity of the solution and
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.
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.
A revised version accepted to Mathematics of Computation, 2001.
Published as a technical report
UCD-CCM 132, 1998, at the
Center for Computational Mathematics, University of Colorado Denver
Viewgraphs of the talk are avalable at
http://math.ucdenver.edu/~aknyazev/research/confs/prism01.ps.gz
