## Preprint 2011-11

## Rodolfo Araya, Gabriel R. Barrenechea, Abner Poza, Frederic Valentin:

### 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 non-linear incompressible Navier-Stokes equations. Stokes problems defined element-wisely drive the construction of the residual-based terms which make the present method stable for the finite element pairs P1/Pl, l=0,1. Numerical upwinding is incorporated through an extra control on the advective derivative and on the residual of the divergence equation. Existence of the discrete solution and uniqueness of a non-singular branch of solutions, 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 post-processing 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 asses the theoretical results and validate the RELP method.

This preprint gave rise to the following definitive publication(s):

**Rodolfo ARAYA, Gabriel R. BARRENECHEA, Abner POZA, Frederic VALENTIN: ***Convergence analysis of a residual local projection finite element method for the Navier-Stokes equations*. SIAM Journal on Numerical Analysis, vol. 50, 2, pp. 669-699, (2012).