Stabilization of numerical formulations for convective-diffussive transport problems

Abstract

[Abstract] Numerical simulation in Fluid Mechanics is an extremely difficult task, wich complexity increases exponentially as the velocity of the fluid becomes higher. In particular, it is know that serious troubles are encountered when the FEM is applied to the resolution of high-advective fluid problems, despite the fact that the method has been successfully applied to a large number of sundry problems of Computational Mechanics. As a general rule, these drawbacks are announced by large oscillations of the Galerkin numerical solution in specific areas, or even through the whole domain. In order to understand the reasons for this unexpected behaviour, we focus our attention in the convective-diffusive transport differential equation, which can be interpreted as the linear version of the Navier-Stokes equations. By means of this simplified analysis, we try to identify the origin of the numerical oscillations phenomena, as much as to find a generic way to stabilize the numerical solution of the problem. In this paper we review the most significant alternative approaches that have been proposed to overcome these troubles when the Galerkin formulation is intended to solve the problem. Then, we propose a new technique that allows to obtain the stabilization parameters for the Petrov-Galerkin approach. Our procedure is baased on the eigenvalue analysis of the elemental matrices of the discretized problem. Thus, the outlined process could be applied independently on the specific formulation being used and the dimension of the problem being solved. Finally, we present different convective-diffusive numerical tests for different Péclet number.