Multilevel preconditioned augmented Lagrangian techniques for 2nd order mixed problems

We are concerned with the efficient solution of saddle point problems arising from the mixed discretization of 2nd order elliptic problems in two dimensions. We consider the mixed discretization of the boundary value problem by means of lowest order Raviart-Thomas elements. This leads to a saddle point problem, which can be tackled by Uzawa-like iterative solvers. We suggest a prior modification of the saddle point problem according to the augmented Lagrangian approach (cf. [16]) in order to make it more amenable to the iterative procedure. In order to boost the speed of iterative methods, we additionally employ a multilevel preconditioner first presented by Vassilevski and Wang in [26]. It is based on a special splitting of the space of vector valued fluxes, which exploits the close relationship between piecewise linear continuous finite element functions and divergence free fluxes. We prove that this splitting gives rise to an optimal preconditioner: it achieves condition numbers bounded independently on the depth of refinement. The proof is set in the framework of Schwarz methods (cf. [28,30]). It relies on established results about standard multilevel methods as well as a strengthened Cauchy-Schwarz inequality for RT₀-spaces.