  In this paper, an efficient parallel splitting method is proposed for the
optimal control problem with parabolic equation constraints. The linear finite
element is used to approximate the state variable and the control variable in
spatial direction. And the Crank-Nicolson scheme is applied to discretize the
constraint equation in temporal direction. For consistency, the trapezoidal
rule and midpoint rule are used to approximate the integrals with respect to
the state variable and the control variable of the objective function in
temporal direction, respectively. Based on the separable structure of the
resulting coupled discretized optimization system, a full Jacobian
decomposition method with correction is adopted to solve the decoupled
subsystems in parallel, which improves the computational efficiency
significantly. Moreover, the global convergence estimate is established using
the discretization error by the finite element and the iteration error by the
full Jacobian decomposition method with correction. Finally, numerical
simulations are carried out to verify the efficiency of the proposed method.
