A posteriori error estimates for the time-dependent convection-diffusion-reaction equation coupled with the Darcy system

In this article, we consider the time-dependent convection-diffusion-reaction equation coupled with the Darcy equation. We propose a numerical scheme based on finite element methods for the discretization in space and the implicit Euler method for the discretization in time. We establish optimal a posteriori error estimates with two types of computable error indicators, the first one linked to the time discretization and the second one to the space discretization. Finally, numerical investigations are performed and presented.