Thesis Summary:

Fluid Mechanics studies the behavior of gases and liquids in motion and is a fundamental tool in disciplines as diverse as Aeronautics, Chemical, Civil and Industrial Engineering, Meteorology, Medicine, Naval Constructions and Oceanography, to name only some. Turbulent air flows or eddies that form when water flows through a pipe or blood through an artery are examples of fluids that appear in these application areas. To describe the motion of a fluid, the so - called Navier - Stokes equations introduced by the French engineer Claude - Louis Navier (1785-1836) and the Irish mathematician George Stokes (1819-1903) are considered. These equations are obtained by applying the principles of mass conservation, or continuity equation, and conservation of moment of inertia, or momentum. As a result, we obtain a set of non-linear partial differential equations, of which an explicit general solution is not available, and apart from certain types of flows and very specific situations, it is not possible to find an analytical solution. To solve this type of problem, several numerical simulation techniques have been developed, which have had an important development in the last decades due to the progressive development of computing power of computers. This methodology of resolution is known as Computational Fluid Dynamics (CFD), where the equations are solved in approximate forms by numerical algorithms that provide information on the values ​​of speed and pressure, among other results of interest. The aim of this thesis is to propose new stabilized finite element schemes for the Navier Stokes stationary equation. For this, new methods of stabilized finite elements are presented along with a posteriori error estimations that allow to improve the solution reached with the numerical method.

ISI Publications from the Thesis

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).

Rodolfo ARAYA, Abner POZA, Frederic VALENTIN: On a hierarchical error estimator combined with a stabilized method for the Navier–Stokes equations. Numerical Methods for Partial Differential Equations, vol. 28, 3, pp. 782–806, (2012).