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Analysis and convergence of a covolume method for the generalized Stokes problem. (English) Zbl 0854.65091
Summary: 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.

##### MSC:
 65N15 Error bounds (BVP of PDE) 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) 76D07 Stokes and related (Oseen, etc.) flows 35B45 A priori estimates for solutions of PDE 35J50 Systems of elliptic equations, variational methods