The Convergence of Iterative Methods for Symmetric and Indefinite Linear Systems

Iterative solution methods provide the only feasible alternative to direct methods for very large scale linear systems such as those which derive from approximation of many partial differential equation problems. For symmetric and positive definite coefficient matrices the conjugate gradient method provides an efficient and popular solver, especially when employed with an appropriate preconditioner. Part of the success of this method is attributable to the rigorous and largely descriptive convergence theory which enables very large sized problems to be tackled with confidence. Here, we describe some convergence results for symmetric and indefinite coefficient matrices which depend on an asymptotically small parameter such as the mesh size in a finite difference or finite element discretization. These estimates are seen to be descriptive in numerical calculations.