Analysis and convergence of a covolume method for the generalized Stokes problem

We introduce a covolume or MAC-like method for approximating the generalized Stokes problem. Two grids are needed in the discretization; a triangular one for the continuity equation and a quadrilateral one for the momentum equation. The velocity is approximated using nonconforming piecewise linears and the pressure piecewise constants. Error in the L(2) norm for the pressure and error in a mesh dependent H-1 norm as well as in the L(2) norm for the velocity are shown to be of first order, provided that the exact velocity is in H-2 and the true pressure in H-1. We also introduce the concept of a network model into the discretized linear system so that an efficient pressure-recovering technique can be used to simplify a great deal the computational work involved in the augmented Lagrangian method. Given is a very general decomposition condition under which this technique is applicable to other fluid problems that can be formulated as a saddle-point problem.