Preconditioned Iterative Methods in a Subspace
               for Linear Algebraic Equations with Large Jumps
                             in the Coefficients

                             Nikolai S. Bakhvalov
                      Institute of Numerical Mathematics
                         Russian Academy of Sciences
                                Moscow, Russia

                              Andrew V. Knyazev
                          Department of Mathematics
                      University of Colorado at Denver 
                               P.O. Box 173364
                                Campus Box 170
                            Denver, CO 80217-3364

                                   Abstract

We consider a family of symmetric matrices $A_\omega=A_0+\omega B$, with a
nonnegative definite matrix $A_0$, a positive definite matrix $B$, and a
nonnegative constant $\omega\le 1$.  Small $\omega$ leads to a poor
conditioned matrix $A_\omega$ with jumps in the coefficients.  For solving
linear algebraic equations with the matrix $A_\omega$, we use standard
preconditioned iterative methods with the matrix $B$ as a preconditioner.  We
show that a proper choice of the initial guess makes possible keeping all
residuals in the subspace IM($A_0$).  Using this property we estimate,
uniformly in $\omega$, the convergence of the methods.

Algebraic equation of this type arise naturally as finite element
discretizations of boundary value problems for PDE with large jumps of
coefficients.  For such problems the rate of convergence does not decrease
when the mesh gets finer and/or $\omega$ tends to zero; each iteration has
only a modest cost.  The case $\omega=0$ corresponds to the fictitious
component/capacitance matrix methods.