Numerical analysis of a stabilized scheme for an optimal control problem governed by a parabolic convection--diffusion equation

We consider an optimal control problem on a bounded domain $\Omega\subset\mathbb{R}^2,$ governed by a parabolic convection--diffusion equation with pointwise control constraints. We follow the optimize--then--discretize--approach, where for the state and the co-state variables, we consider the piecewise finite element method alongside the algebraic flux correction method for its stabilization and the for temporal discretization, we use the backward Euler method for the state variable and the explicit Euler method for the co-state variable. The discrete control variable is obtained by projecting the discretized adjoint state onto the set of admissible controls. The resulting stabilized fully--discrete scheme is nonlinear and a fixed point argument is used in order to prove its existence and uniqueness under a mild condition between the time step $k$ and the mesh step $h,$ e.g., $k = \mathcal{O}(h^{1+\epsilon}),\,0<\epsilon<1.$ Further, for sufficiently regular solution, we derive error estimates in $L^2$ and $H^1$ norm with respect on space and $\ell^\infty$ norm in time for the state and the co-state variables. For the control variable we also derive an $L^2$ estimate for its error with respect to spatial variable and $\ell^\infty$ in time. Finally, we present numerical experiments that validate the the order of convergence of the stabilized fully--discrete scheme via the algebraic flux correction method as well as we test the stabilized fully--discrete scheme in optimal control problems governed by a convection--dominant equation where the solution possesses interior or boundary layers.