Convergence Analysis of a Residual Local Projection Finite Element Method for the Navier–Stokes Equations

Abstract

This work presents and analyzes a new residual local projection stabilized finite element method (RELP) for the nonlinear incompressible Navier-Stokes equations. Stokes problems defined elementwise drive the construction of the residual-based terms which make the present method stable for the finite element pairs $\mathbb{P}_1/\mathbb{P}_l$, $l=0,1$. Numerical upwinding is incorporated through an extra control on the advective derivative and on the residual of the divergence equation. Well-posedness of the discrete problem as well as optimal error estimates in natural norms are proved under standard assumptions. Next, a divergence-free velocity field is provided by a simple postprocessing of the computed velocity and pressure using the lowest order Raviart-Thomas basis functions. This updated velocity is proved to converge optimally to the exact solution. Numerics assess the theoretical results and validate the RELP method.