Elliptic problems with highly discontinuous leading coefficients appear in many applications, e.g., in homogenization of composites with a periodic structure. 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. This leads to natural difficulties such as poor approximation of the solution by traditional methods and slow convergence of standard iterative algorithms.

However, we suggest a natural implicit splitting of the problem into two well-posed problems. Using this idea, we prove a uniform (with respect to the jump in the coefficients) convergence of a standard preconditioned iterative method with a special initial guess. We also demonstrate uniform finite element method error estimates for such problems. In all our arguments we use a natural parameter-independent Sobolev norm, not the energy norm.

The talk is aimed for a general applied math audience and is based on the following papers:

N. S. Bakhvalov, A. V. Knyazev, and R. R. Parashkevov, Extension Theorems for Stokes and Lame equations for nearly incompressible media and their applications to numerical solution of problems with highly discontinuous coefficients. Numerical Linear Algebra with Applications, 9 (2002) no. 2, 115-139.
A. V. Knyazev and Olof Widlund, Lavrentiev Regularization + Ritz Approximation = Uniform Finite Element Error Estimates for Differential Equations with Rough Coefficients. Mathematics of Computation, posted on July 13, 2001, S0025-5718-01-01378-3 (to appear in print).
N. S. Bakhvalov and A. V. Knyazev, Preconditioned Iterative Methods in a Subspace, In Domain Decomposition Methods in Science and Engineering, Ed. D. Keyes and J. Xu, AMS, 157-162, 1995.

N. S. Bakhvalov and A. V. Knyazev, Fictitious domain methods and computation of homogenized properties of composites with a periodic structure of essentially different components, In Numerical Methods and Applications, Ed. Gury I. Marchuk, CRC Press, 221-276, 1994.