Galerkin, Least-Squares and GLS numerical approaches for advective-diffussive transport problems in engineering

European Congress on Computational Methods in Applied Sciences and Engineering, Barcelona 11-14 september 2000[Abstract] In this paper, a study of three FE numerical formulations (Galerkin, Least Squares and Galerkin/Least Squares) applied to the convective-diffuse problem is presented, focusing our attention in high Péclet-number problems. The election of these three approaches is not arbitrary, but based on the relations among them. First, we review the causes of appearance of numerical oscillations when a Galerkin formulation is used. Contrasting with the nature of the Galerkin method, the Least Squares methos has a rigorous foundation on the basis of minimizing the square residual, which ensures best numerical results. However, this improvement in the numerical solution implies an increment of the computational cost, wich normally becomes unaffordable in practice. The last one, know as GLS, is based on a stabilization of the Galerkin Method. GLS can be interpreted as a combination of the last two methods, being one to solve convective problems, because it unifies the advantages of the Galerkin and Least Squares Methods and cancels its disadventages. For each numerical method, a brief review is presented, the continuity and derivability requirements on the trial functions are stablished, and the reasons of its behavior when the method is applied to the convection-diffusion problem with high velocity fields are examined. Furthermore, special attention will be devoted to the consequences of relaxing the variational requirements in the LS and GLS methods. Finally, several 1D and 2D examples are presented

Petrov-Galerkin and Galerkin/Least Squares stabilized approaches for advectivediffusive transport problems

[Abstract] The Finite Element method with a Galerkin type weighting is a straight-forward weighted residual method that has been sucessfuly used in many engineering applications, specially in Solid Mechanics. However, this method yields oscillatory solutions when it is applied to high-advective problems in Fluid Mechanics. Several stabilized numerical formulations have been proposed in the last years to overcome these inestabilities. The common methodology of most of these approaches is based on the addition of a term to the Galerkin formulation, in order to enhance the estability behaviour while preserving the weighting residual scheme. In this paper, we focus our attention in the Stream Upwind/Petrov Galerkin method (SUPG), and the Galerkin/least-squares method (GLS). We will review the mathematical formulation of both of them, as well as the key concept of their respective fundamentals and derivations, i.e. the exact artificial diffusion method for the SUPG and the Least Squares Finite Element method for the GLS. Finally, we will present a comparision between both methods, pointing out important coincidences and estabishing their mutual relations.Xunta de Galicia; PGDIT99MAR1180