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

In this paper we study the a posteriori error estimates for the time dependent Navier-Stokes system coupled with the convection-diffusion-reaction equation. The problem is discretized in time using the implicit Euler method and in space using the finite element method. We establish a posteriori error estimates with two types of computable error indicators, the first one linked to the space discretization and the second one to the time discretization. Finally, numerical investigations are performed and presented.