Preconditioning in H$(\operatorname {div})$ and applications

We consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator I$- \operatorname {\mathbf {grad}}\operatorname {div}$. The natural setting for such problems is in the Hilbert space H$(\operatorname {div})$ and the variational formulation is based on the inner product in H$(\operatorname {div})$. We show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques. These preconditioners are shown to be spectrally equivalent to the inverse of the operator. As a consequence, they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations, with the number independent of the mesh discretization. We describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems.