A PETROV-GALERKIN FORMULATION FOR ADVECTION-REACTION-DIFFUSION PROBLEMS

In this work we present a new method called (SU + C)PG to solve advect ion-reaction-diffusion scalar equations by the Finite Element Method ( FEM). The SUPG (for Streamline Upwind Petrev-Galerkin) method is curre ntly one of the most popular methods for advection-diffusion problems due to its inherent consistency and efficiency in avoiding the spuriou s oscillations obtained from the plain Galerkin method when there are discontinuities in the solution. Following this ideas, Tezduyar and Pa rk treated the more general advection-reaction-diffusion problem and t hey developed a stabilizing term for advection-reaction problems witho ut significant diffusive boundary layers. In this work an SUPG extensi on for all situations is performed, covering the whole plane represent ed by the Peclet number and the dimensionless reaction number. The sch eme is based on the extension of the super-convergence feature through the inclusion of an additional perturbation function and a correspond ing proportionality constant. Both proportionality constants (that one corresponding to the standard perturbation function from SUPG, and th e new one introduced here) are selected in order to verify the 'super- convergence' feature, i.e. exact nodal values are obtained for a restr icted class of problems (uniform mesh, no source term, constant physic al properties). It is also shown that the (SU + C)PG scheme verifies t he Discrete Maximum Principle (DMP), that guarantees uniform convergen ce of the finite element solution. Moreover, it is shown that super-co nvergence is closely related to the DMP, motivating the interest in de veloping numerical schemes that extend the super-convergence feature t o a broader class of problems.