A Posteriori Error Estimates for Unsteady Convection-Diffusion-Reaction Problems and the Finite Volume Method

We derive a posteriori error estimates for the discretization of the unsteady linear convection-diffusion-reaction equation approximated with the cell-centered finite volume method in space and the backward Euler scheme in time. The estimates are based on a locally postprocessed approximate solution preserving the conservative fluxes and are established in the energy norm. We propose an adaptive algorithm which ensures the control of the total error with respect to a user-defined relative precision and refines the meshes adaptively while equilibrating the time and space contributions to the error. Numerical experiments illustrate the theory.