Preconditioning Mixed Finite Element Saddle-point Elliptic Problems

We consider saddle-point problems that typically arise from the mixed finite element discretization of second order elliptic problems. By proper equivalent algebraic operations the considered saddle-point problem is transformed to another saddle-point problem. The resulting problem can then be efficiently preconditioned by a block-diagonal matrix or by a factored block-matrix (the blocks correspond to the velocity and pressure, respectively). Both preconditioners have a block on the main diagonal that corresponds to the bilinear form R Ω a 1 + 1 r r ( is a positive parameter) and a second block that is equal to a constant times the identity operator. We derive uniform bounds for the negative and positive eigenvalues of the preconditioned operator. Then any known preconditionerfor the above bilinear form can be applied. We also showsome numerical experiments that illustrate the convergence properties of the proposed technique.