### FINITE ELEMENT METHODS FOR THE NUMERICAL SIMULATION OF INCOMPRESSIBLE VISCOUS FLUID FLOW MODELED BY THE NAVIER-STOKES EQUATIONS. PART I

*R. Glowinski, T. W. Pan, L. H. Juárez, E. Dean*

#### Resumen

The Navier–Stokes equations have been known for more than a century and they still provide the most commonly used mathematical model to describe and study the motion of viscous fluids, including phenomena as complicated as turbulent flow. One can only marvel at the fact that these equations accurately describe phenomena whose length scales (resp., time scale) range from fractions of a millimeter (resp., of a second) to thousands of kilometers (resp., several years). Indeed, the Navier–Stokes equations have been validated by numerous comparisons between analytical or computational results and experimental measurements; some of these comparisons are reported in Canuto et al. 1988 [1], Lesieur 1990 [2], Guyon et al. 1991 [3], and Glowinski 2003 [4].

This note does not have the pretension to cover the full field of finite element methods for the Navier–Stokes equations and is organized in sections as follows:

1. The Navier–Stokes equations for incompressible viscous flow

2. Some operator splitting methods for initial value problems and applications to the Navier–Stokes equations

3. Iterative solution of the advection–diffusion sub–problems and the wavelike equation method for the advection sub–problems

4. Iterative solution of the Stokes type sub–problem

5. Finite element approximation of the Navier–Stokes equations

6. Numerical results

This note does not have the pretension to cover the full field of finite element methods for the Navier–Stokes equations and is organized in sections as follows:

1. The Navier–Stokes equations for incompressible viscous flow

2. Some operator splitting methods for initial value problems and applications to the Navier–Stokes equations

3. Iterative solution of the advection–diffusion sub–problems and the wavelike equation method for the advection sub–problems

4. Iterative solution of the Stokes type sub–problem

5. Finite element approximation of the Navier–Stokes equations

6. Numerical results

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