A priori error estimates of a discontinuous Galerkin method for the Navier-Stokes equations

This-36pt paper considers a discontinuous Galerkin finite element method for the 2 D transient and incompressible Navier-Stokes model. Following the analysis of Heywood and Rannacher ( SIAM J. Numer. Anal. 19:275–311, 1982), we derive optimal velocity and pressure error estimates in L ∞ ( L 2 ) and L ∞ ( L 2 ) -norms, respectively, for the discontinuous Galerkin case. We use standard L 2 -projection and modified Stokes operator but on appropriate broken Sobolev spaces, and then standard duality arguments to achieve these results. For sufficiently small data, uniform in time estimates are proved. Based on the backward Euler method, time discretization is carried out and fully discrete error estimates are derived. Finally, numerical experiments are conducted to verify our theoretical findings.