A Schwarz Preconditioner for a Hybridized Mixed Method

In this paper, we provide a Schwarz preconditioner for the hybridized versions of the Raviart-Thomas and Brezzi-Douglas-Marini mixed methods. The preconditioner is for the linear equation for Lagrange multipliers arrived at by eliminating the flux as well as the primal variable. We also prove a condition number estimate for this equation when no preconditioner is used. Although preconditioners for the lowest-order case of the Raviart-Thomas method have been constructed previously by exploiting its connection with a nonconforming method, our approach is different in that we use a new variational characterization of the Lagrange multiplier equation. This allows us to precondition even the higher-order cases of these methods. 2000 Mathematics Subject Classification: 65N55, 65N22, 65N30.