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

Authors:
Rodolfo Araya;Gabriel R. Barrenechea;Abner H. Poza;Frédéric Valentin
Affiliations:
raraya@ing-mat.udec.cl;gabriel.barrenechea@strath.ac.uk;abner@ing-mat.udec.cl;valentin@lncc.br
Venue:
SIAM Journal on Numerical Analysis
Year:
2012

Citing 6
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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.