(A) conservative covolume method for the navier-stokes equations

Abstract

We introduce a new covolume method for approximating the stationary Navier-Stokes equations and analyze its convergence. There are two ways to introduce the covolume approximation to the Navier-Stokes equations. One uses the divergence (or conservative) form of Navier-Stokes equations which we call the conservative covolume method, the another uses its original form. Primal and dual grids are used in the covolume method. Test functions are piecewise constant on the dual grid. In the covolume method the momentum equation is integrated over the dual element and the continuity equation over the primal element. The finite element space for the velocity is the Crouzeix-Raviart space for triangles or nonconforming $P_1$ element consisting of piecewise linear functions and the finite element space for the pressure is the space of piecewise constant functions on the primal elements, whereas the test function space for the velocity consists of certain piecewise constant functions on the dual elements. An abstract theory based on the results of approximation for branches of nonsingular solutions of nonlinear problems gives us an opportunity to study of the convergence of the covolume method for the Navier-Stokes equations. Efficiency of the proposed method has been tested on a number of test problems. Numerical results using a simple Picard type of iteration for solving the discrete Navier-Stokes equations are provided.