A high order stabilized finite element formulation for scalar convection-diffusion problems.∗ Eduardo G. D. do Carmo, Department of Nuclear Engineering, COPPE , UFRJ, 21945-970, P. B. 68509, Ilha do Fund˜ ao, Rio de Janeiro, RJ E-mail:
[email protected],
Gustavo B. Alvarez†, Abimael F. D. Loula LNCC - National Laboratory of Scientific Computation Av. Get´ ulio Vargas, 333 25651-070, P.B. 95113, Petr´ opolis, RJ, E-mail:
[email protected],
In general, the solution of diffusive-convective problems possesses boundary layers where derivatives of the solution are very large. It is well known that Galerkin finite element method is instable for this class of problems. Stabilized finite element methods such as SUPG [1] (Streamline Upwind/Petrov-Galerkin) and GLS [4] (GalerkinLeast-Squares) have been proposed as alternatives to the Galerkin formulation. These methods add stabilization terms to the Galerkin formulation, providing good stability and accuracy to the numerical solution, when the exact solution of the problem is smooth. However, the spurious oscillations remain when the exact solution possesses boundary layers. Many attempts have used the SUPG or GLS methods as a starting point for new stabilized formulations. In particular, the CAU method [3] (Consistent Approximate Upwind) maintains the term of SUPG and adds a nonlinear term that eliminates the spurious oscillations. Nevertheless, when the exact solution of the problem is smooth, the CAU’s approximate solution presents undesirable crosswind diffusion. This paper presents an accurate stabilized finite element formulation for convection dominated problems. The new method uses the GLS formulation as starting point. The paper presents a study on the function of the P´eclet number that appears in the upwind function of the GLS, CAU and SAUPG [2] (Streamline and Approximate Upwind/PetrovGalerkin) methods, using linear and quadratic triangular elements, as well as bilinear and biquadratic quadrilateral elements. The numerical experiments indicate that in the case of elements with faces on the ‘outflow boundary´, the upwind function is strongly dependent on the geometry and on ∗ The authors wish to thank the Brazilian research funding agencies CNPq and FAPERJ for its support to this work. † bolsista de Fixa¸ c˜ ao de Pesquisador FAPERJ
[email protected]
the degree of the polynomial. A new function of the P´eclet number is proposed, to take these effects into consideration, creating a new method of stabilization for higher order elements.
Referˆ encias [1] A. N. Brooks, T. J. R. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 32 (1982) 199-259. [2] E. G. D. do Carmo and G. B. Alvarez, A new stabilized finite element formulation for scalar convection-diffusion problems: the streamline and approximate upwind/Petrov-Galerkin method, Comput. Methods Appl. Mech. Engrg. 192 (2003) 3379-3396. [3] A. C. Gale˜ ao and E. G. Do Carmo, A consistent approximate upwind Petrov-Galerkin method for convection-dominated problems, Comput. Methods Appl. Mech. Engrg. 68 (1988) 83-95. [4] T. J. R. Hughes, L. P. Franca, G. M. Hulbert, A new finite element formulation for computational fluid dynamics: VII. The GalerkinLeast-Squares method for advective-diffusive equations, Comput. Methods Appl. Mech. Engrg. 73 (1989) 173-189.
