In this article, we consider the convection-diffusion-reaction problem coupled the Darcy-Forchheimer problem by a non-linear external force depending on the concentration. We establish existence of a solution by using a Galerkin method and we prove uniqueness. We introduce and analyse a numerical scheme based on the finite element method. An optimal a priori error estimate is then derived for each numerical scheme. Numerical investigation are performed to confirm  the theoretical accuracy of the discretization.

This work deals with the a posteriori error estimates for the Darcy–Forchheimer problem. We first introduce the corresponding variational formulation for the nonlinear problem and discretize it by using the finite-element method. We then propose a linear iterative scheme to solve the nonlinear variational problem for a fixed mesh step. Finally, a posteriori error estimate with two types of computable error indicators is showed. The first one is linked to the linearization and the second one to the discretization. Numerical computations are performed to show the effectiveness of the derived error indicators.