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

We consider a family of symmetric matrices Aω = A0 +ωB, with a nonnegative definite matrix A0, a positive definite matrix B, and a nonnegative parameter ω ≤ 1. Small ω leads to a poor conditioned matrix Aω with jumps in the coefficients. For solving linear algebraic equations with the matrix Aω, 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(A0). Using this property we estimate, uniformly in ω, the convergence rate of the methods. Algebraic equations 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 ω tends to zero; each iteration has only a modest cost. The case ω = 0 corresponds to the fictitious component/capacitance matrix method.