Fast solvers for mixed finite element methods

Discretizations of partial differential equations by mixed finite element methods result in large saddle point systems that require efficient solvers. In this chapter we present two classes of iterative methods for such problems: the (preconditioned) Uzawa algorithm and the (preconditioned) minimum residual algorithm. The implementations of these algorithms require either efficient preconditioners for the discrete Laplace operator and/or the efficient solution of the discrete Poisson problem. Towards this end we provide brief introductions to additive Schwarz (domain decomposition/multilevel) preconditioners and multigrid algorithms, after a discussion on block diagonal preconditioners for saddle point problems