(SU+C)PG: A Petrov-Galerkin Formulation for Advection-Reaction-Diffusion Problems

Abstract

In this work we present a new method called (SU+C)PG to solve advection-react iondiffusion (ARD) scalar equations by the Finite Element Method (FEM) [lJ. Following the ideas behind SUPG [2, 3], Tezduyar and Park treated the more general ARD problem and they developed a stabilizing term for advection-reaction problems without significant diffusive boundary layers. In this work a PG extension for all situations is performed, covering the whole plane represented by the Peclet number and the dimensionless
reaction number. The scheme is based in extending the super-convergence feature through the inclusion of an additional perturbation function and a corresponding
proportionality constant. The proportionality constants are selected in order to verify the "super-convergence" feature. i.e. exact nodal values are obtained for a restricted
class of problems (uniform mesh, no source term, constant physical properties).
It is also shown that the (SU+C)PG scheme verifies the Discrete Maximum Principle (DMP), that guarantees uniform convergence of the finite element solution. Moreover, it is shown that super-convergence is closely related to the DMP, motivating the interest in developing numerical schemes that extend the super-convergence feature to a broader class of problems.