Preconditioned Iterative Methods in a Subspace

We consider a family of symmetric matrices A! = A 0 + !B; with a nonnegative deenite matrix A 0 ; a positive deenite matrix B; and a nonnegative parameter ! 1: Small ! leads to a poor conditioned matrix A! with jumps in the coeecients. For solving linear algebraic equations with the matrix A!; 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 !; the convergence rate of the methods. Algebraic equations of this type arise naturally as nite element dis-cretizations of boundary value problems for PDE with large jumps of coef-cients. For such problems the rate of convergence does not decrease when the mesh gets ner and/or ! tends to zero; each iteration has only a modest cost. The case ! = 0 corresponds to the ctitious component/capacitance matrix method.