A non-conforming finite volume element method of weighted upstream type for the two-dimensional stationary Navier–Stokes system

We consider the discretization of the stationary Navier–Stokes system in a two-dimensional domain by a non-conforming finite volume element method. We use the standard formulation of the Navier–Stokes system in the primitive variables and take as approximation space the non-conforming P1-elements for the velocity and piecewise constant elements for the pressure. The non-linear convective term is treated using an upstream approach with weight, based on the scheme from [F. Schieweck, L. Tobiska, A non-conforming finite element method of upstream type applied to the stationary Navier–Stokes equation, M2AN 23 (1989) 627–647]. For the proposed scheme, we prove existence and uniqueness results (under the standard assumption that the datum has to be sufficiently small with respect to the viscosity parameter, cf. [R. Temam, Navier–Stokes Equations, North-Holland, Amsterdam, 1984]). An error estimate in the energy norm is proved and is confirmed by different numerical tests.