Finite volume element discretization of optimal control of the parabolic equation using the discretize-then-optimize approach

This paper proposes a novel finite volume element (FVE) scheme for linear parabolic optimal control problems (OCPs) subject to integral control constraints. The state and co-state variables were approximated using continuous piecewise linear finite elements, while the control variable was discretized via piecewise constant functions. First, following the discretize-then-optimize approach, the FVE approximation of the parabolic OCP was formulated. Second, the first-order optimality conditions were derived, and corresponding error estimates in the $L^2(J;H^1(\mathrm{\Omega }))$-norm for the state and co-state variables, as well as in the $L^2(J;L^2(\mathrm{\Omega }))$-norm for the control variable, were established. These estimates quantify the deviation between the discrete solutions and the exact solutions over the time interval $J$ and spatial domain $\mathrm{\Omega }$, providing rigorous bounds on the approximation errors. Third, some superclose results between the projection of the exact solution and the discrete solution for all variables were analyzed, leading to optimal-order error estimates in the $L^{\mathrm{\infty }}(J;L^2(\mathrm{\Omega }))$-norm for all variables. Finally, a numerical example was presented to validate the theoretical results. We believe that this is the first article to construct an FVE approximation based on the discretize-then-optimize approach for the parabolic OCP.