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 L2 norm for the pressure and error in a mesh dependent H1 norm as well as in the L2 norm for the velocity are shown to be of first order, provided that the exact velocity is in H2 and the true pressure in H1. 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.