Numerical behavior of saddle point solvers.

Symmetric indefinite saddle-point problems arise in many application areas such as computational fluid dynamics, electromagnetism, optimization and nonlinear programming. Particular attention has been paid to their iterative solution. In this talk we analyze several theoretical issues and practical aspects related to the application of preconditioners in Krylov subspace methods. Several structure--dependent schemes have been proposed and analyzed. Indeed, the nature of these systems enables to take into account not only simple preconditioning strategies and scalings, but also preconditioners with a particular block structure. It is well-known that the application of positive definite block-diagonal preconditioner still leads to preconditioned system with a symmetric structure similar to the original saddle point system. On the other hand, the application of symmetric indefinite or nonsymmetric block-triangular preconditioner leads to nonsymmetric triangular preconditioned systems and therefore general nonsymmetric iterative solvers should be considered. The experiments however indicate that Krylov subspace methods perform surprisingly well on practical problems even those which should theoretically work only for symmetric systems.