A priori error estimates of finite volume element method for bilinear parabolic optimal control problem

In this paper, we study the finite volume element method of bilinear parabolic optimal control problem. We will use the optimize-then-discretize approach to obtain the semi-discrete finite volume element scheme for the optimal control problem. Under some reasonable assumptions, we derive the optimal order error estimates in $L^2(J;L^2)$ and $L^{\mathrm{\infty }}(J;L^2)$-norm. We use the backward Euler method for the discretization of time to get fully discrete finite volume element scheme for the optimal control problem, and obtain some error estimates. The approximate order for the state, costate and control variables is $O(h^{3/2}+\mathrm{△}t)$ in the sense of $L^2(J;L^2)$ and $L^{\mathrm{\infty }}(J;L^2)$-norm. Finally, a numerical experiment is presented to test these theoretical results.