Convergence and stability of a stabilized finite volume method for the stationary Navier-Stokes equations

In this paper, a new stabilized finite volume method is studied and developed for the stationary Navier-Stokes equations. This method is based on a local Gauss integration technique and uses the lowest equal order finite element pair P1–P1 (linear functions). Stability and convergence of the optimal order in the H1-norm for velocity and the L2-norm for pressure are obtained. A new duality for the Navier-Stokes equations is introduced to establish the convergence of the optimal order in the L2-norm for velocity. Moreover, superconvergence between the conforming mixed finite element solution and the finite volume solution using the same finite element pair is derived. Numerical results are shown to support the developed convergence theory.