An analysis of a weak Galerkin finite element method for stationary Navier–Stokes problems

Abstract

In this article, we present and analyze a weak Galerkin finite element method for stationary Navier–Stokes problems. This weak Galerkin finite element scheme is based on a shape regular partition consisting of arbitrary polygons/polyhedra. We first establish a discrete embedding inequality that is useful in weak Galerkin finite element analysis for nonlinear problems. Then the stability and unique existence are proved for the discrete velocity and pressure by means of a discrete inf–sup condition. Furthermore, we derive the optimal error estimates for velocity approximation in the discrete $H^1$-norm and pressure approximation in the $L_2$-norm, respectively. Numerical examples are provided that corroborate the optimal convergence of the proposed method.