A Preconditioned Gmres Method for Nonsymmetric or Indefinite Problems

A preconditioning technique is proposed for nonsymmetric or indefinite linear systems of equations. The main idea in our theory, roughly speaking, is first to use some "coarser mesh" space to correct the nonpositive portion of the eigenvalues of the underlying operator and then switch to use a symmetric positive definite preconditioner. The generality of our theory allows us to apply any known preconditioners that were orginally designed for symmetric positive definite problems to nonsymmetric or indefinite problems, without losing the optimality that the original one has. Some numerical experiments based on GMRES are reported.