Numerical solution of elliptic problems with highly discontinuous coefficient

Prof. Andrew Knyazev

Department of Mathematics, University of Colorado at Denver, Denver, CO 80217-3364

We consider, as an example, 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, 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.