A Petrov-Galerkin formulation for advection-reaction-diffusion problems (Q1371824)

We present a method called \((\text{SU}+\text{C})\text{PG}\) to solve advection-reaction-diffusion scalar equations by the finite element method. The SUPG (streamline upwind Petrov-Galerkin) extension is performed, covering the whole plane represented by the Péclet number and the dimensionless reaction number. The scheme is based on the extension of the super-convergence feature through the inclusion of an additional perturbation function and a corresponding proportionality constant. Both proportionality constants (the one corresponding to the standard perturbation function from SUPG, and the new one introduced here) are selected in order to verify the super-convergence. It is also shown that the \((\text{SU}+\text{C})\text{PG}\) scheme satisfies the discrete maximum principle, that guarantees uniform convergence of the finite element solution.