Fully Discrete Finite Element Method Based on the Crank-Nicolson Scheme for Parabolic Optimal Control Problem

Aiming at parabolic optimal control problem with integral control constraints, a fully discrete finite element method based on Crank-Nicolson scheme is proposed. The state is discretized by piecewise linear finite elements for the space discretization, Crank-Nicolson scheme for time discretization, and the control variables are approximated by piecewise linear function approximation, so as to obtain a discrete optimal system. The error estimates of state variables, adjoint state variables and control variables are proved, and the theoretical results are verified by numerical examples.