@article{Hou2026, 
author = {Chunjuan Hou and Baitong Ma},
title = {Finite volume element discretization of optimal control of the parabolic equation using the discretize-then-optimize approach},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {1},
pages = {444-461},
keywords = {parabolic problem, optimal control, finite volume, convergence, superclose},
url = {https://www.sciopen.com/article/10.3934/math.2026019},
doi = {10.3934/math.2026019},
abstract = {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    (  Ω  )  )-norm for the state and co-state variables, as well as in the        L    2    (  J  ;      L    2    (  Ω  )  )-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    Ω, 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    ∞    (  J  ;      L    2    (  Ω  )  )-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.}
}