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.

This paper addresses the numerical method for the generalized Allen–Cahn equation featuring nonlinear mobility and a convection term. We propose a linear second–order finite difference scheme that adheres to the discrete maximum bound principle (MBP). The scheme is discretized using the leapfrog finite difference approach, incorporating a stabilized term in time, an upwind scheme for the convection term, and a central–difference scheme for the diffusion term. It is demonstrated that the discrete MBP holds under reasonable constraints on both the time step size and the coefficient of the stabilized term. Additionally, we provide an $L^{\mathrm{\infty }}\text{–error}$ estimate for our proposed scheme. Several numerical experiments are conducted to validate our theoretical findings.