WWW: 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 Knyazev
 
Numerical Solution of Elliptic Problems with Highly Discontinuous Coefficients Typical for Homogenization

 
Abstract

We consider, as an example, a parametric family of periodic boundary value problems for the diffusion equation with the diffusion coefficient equal to a small constant in a subdomain. Such problems typically appear in the process of homogenization of composites with a periodic structure. They are not uniformly well-posed when the constant gets small. However, we suggest a natural implicit splitting of the problem into two well-posed problems. Using this idea, we prove a uniform convergence of a standard preconditioned iterative method with a special initial guess. In all our arguments we use a natural parameter-independent Sobolev norm, not the energy norm. We also discuss FEM error estimates for such problems, and mention some results for the mixed formulation of the problem.
 
  Schlumberger-Doll Research, Ridgefield, CT, January 26, 1998