Covolume methods for the Navier-Stokes problem

Abstract

We extend recent results of covolume methods for the generalized Stokes problem to a stabilized problem and the stationary incompressible Navier-Stokes problem. Two partitions of the problem domain are needed, the primal partition and the dual partition. It turns out that the covolume methods in this thesis can be viewed as Petrov-Galerkin methods. Hence, some results of finite element methods are used in the analysis of the covolume methods. A stabilized covolume method for the Stokes problem can be formulated by modifying the divergence free condition. We prove that this scheme has a unique solution without the inf-sup condition and its linear convergence in $H^1$ semi-norm for the velocity and in $L^2$ norm for the pressure. We also present numerical results corresponding to this analysis. A major difficulty in numerical methods for the Navier-Stokes problem is in discretizing the convection term. We introduce a skew-symmetric form of the convection term in the problem of small data. We prove that the covolume method for the Navier-Stokes problem has a unique solution and its linear convergence in $H^1$ semi-norm for the velocity and in $L^2$ norm for the pressure. We also consider a multigrid algorithm for the cell centered finite difference scheme on triangular meshes. The energy norm of this prolongation operator is shown to be less than $\sqrt{2}$. Thus the W-cycle is guaranteed to converge. Numerical experiments show that the new prolongation operator is better than the trivial injection.