  This paper concerns robust numerical treatment of an elliptic PDE with high
contrast coefficients, for which classical finite-element discretizations yield
ill-conditioned linear systems. This paper introduces a procedure by which the
discrete system obtained from a linear finite element discretization of the
given continuum problem is converted into an equivalent linear system of the
saddle point type. Then three preconditioned iterative procedures --
preconditioned Uzawa, preconditioned Lanczos, and PCG for the square of the
matrix -- are discussed for a special type of the application, namely, highly
conducting particles distributed in the domain. Robust preconditioners for
solving the derived saddle point problem are proposed and investigated.
Robustness with respect to the contrast parameter and the discretization scale
is also justified. Numerical examples support theoretical results and
demonstrate independence of the number of iterations of the proposed iterative
schemes on the contrast in parameters of the problem and the mesh size.
