Hybridization of mixed methods for the Dirichlet problem by introduction of Lagrange multipliers is the preferred way of implementing the mixed method for compelling theoretical and practical reasons. We introduce a new variational characterization of the Lagrange multiplier equation arising from hybridization. This result has several applications, but its applications in constructing preconditioners will be our main focus. We will begin by examining the conditioning of this equation when no preconditioner is used. Then we will establish certain spectral equivalences which allow construction of a Schwarz preconditioner for the Lagrange multiplier equation. Although preconditioners for the lowest order case of the hybridized Raviart-Thomas method have been constructed previously by exploiting its connection with a nonconforming method, our approach is different in that we use the new variational characterization. This allows us to precondition even the higher order cases of these methods. Among other applications of the characterization result, is a previously unsuspected relationship between the Raviart-Thomas and the Brezzi-Douglas-Marini methods, e.g., we show that when these methods are used to approximate harmonic solutions, they yield identical Lagrange multipliers.

[1] B. Cockburn and J. Gopalakrishnan, A characterization of hybridized mixed methods for the Dirichlet problem. (Preprint.)
[2] J. Gopalakrishnan, A Schwarz preconditioner for a hybridized mixed method. (Preprint.)