A priori error estimates of finite volume method for nonlinear optimal control problem

In this paper, we study a priori error estimates for a finite volume element approximation of a nonlinear optimal control problem. The schemes use discretizations base on a finite volume method. For the variational inequality, we use a method of the variational discretization concept to obtain the control. Under some reasonable assumptions, we obtain some optimal order error estimates. The approximate order for the state, costate, and control variables is $O(h^2)$ or $O(h^2\sqrt{|\ln h|})$ in the sense of $L^2$-norm or $L^\infty$-norm. A numerical experiment is presented to test the theoretical results. Finally, we give some conclusions and future works.