Saddle Point Least Squares Preconditioning of Mixed Methods

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