Abstract

The main goal of this paper is to study adaptive mesh techniques, using a posteriori error estimates, for the finite element solution of the Navier-Stokes equations modeling steady and unsteady flows of viscous fluids. Among existing operator splitting techniques, the $\theta$-scheme is used for time integration of the Navier-Stokes equations. Then a posteriori error estimates, based on the solution of a local system for each triangular element, are derived in the framework of the generalized Stokes problem. Adaptive strategies, including hierarchical and non-hierarchical refinements, and also enhanced by a moving mesh procedure, are developed to implement this mathematical criterion. Numerical simulations of viscous flows of industrial interest around aerodynamic shapes are presented and discussed to demonstrate the accuracy and efficiency of our methodology.