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We present a simple way to discretize and precondition mixed variationalformulations. Our theory connects with, and takes advantage of, the classicaltheory of symmetric saddle point problems and the theory of preconditioningsymmetric positive definite operators. Efficient iterative processes forsolving the discrete mixed formulations are proposed and choices for discretespaces that are always compatible are provided. For the proposed discretespaces and solvers, a basis is needed only for the test spaces and assembly ofa global saddle point system is avoided. We prove sharp approximationproperties for the discretization and iteration errors and also provide a sharpestimate for the convergence rate of the proposed algorithm in terms of thecondition number of the elliptic preconditioner and the discrete $\inf-\sup$and $\sup-\sup$ constants of the pair of discrete spaces.

Saddle Point Least Squares Preconditioning of Mixed Methods