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

Nikolai S. Bakhvalov
Institute of Numerical Mathematics
Moscow, Russia

Andrew V. Knyazev
Department of Mathematics
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.