CONVERGENCE OF SOME ITERATIVE METHODS FOR SYMMETRIC SADDLE POINT LINEAR SYSTEMS

We consider the iterative solution of linear systems with a symmetric saddle point system matrix. We address a family of solution techniques that exploit the knowledge of a preconditioner (or approximate solution procedure) both for the top left block of the matrix on the one hand and for the Schur complement resulting from its elimination on the other hand. This includes many "segregated" or "Schur complement" iterations such as the inexact Uzawa method and pressure correction techniques, and also many "block" preconditioners, based on the approximate block factorization of the system matrix. An analysis is developed which proves convergence in norm of stationary iterations. It is more rigorous than eigenvalue analyses which ignore nonnormality effects, while being more general than previous norm analyses. The analysis also clarifies the relations that exist between the many members of this family of methods and offers practical guidelines to select the scheme most appropriate to a situation at hand.