Convergent Analysis Of Finite Volume Element Method For Optimal Control Problems

The optimal control problem is a optimization problem under the constraints of diferentialequations. As the diferential equations do, the optimal control problem is widely used in oursociety, such as air pollution control, cancer chemotherapy, financial investment, fluid control etc.Finite volume element method is a method with conservation (mass, energy and momentum). It isparticularly suitable for the numerical approximation with conservation issues, such as fluid flowproblems. In this paper we study the convergent analysis of finite volume element method foroptimal control problems. Though we study the finite volume element approximation of severalbasic classes optimal control problems, it is the foundation for further studying the finite volumeelement approximation of some other optimal control problems.In this paper, we apply the optimize-then-discretize technique and the variational discretiza-tion approach to study the finite volume element approximation of the elliptic optimal control,parabolic optimal control and second order hyperbolic optimal control problems. We obtain thecorresponding discrete optimal system, verify the existence and uniqueness of these systems, andderive some convergent results (i.e. some a prior error estimates) for the numerical solution. Forthe evolutional equations optimal problems, we also develop some fully-discrete schemes and getthe corresponding a priori error estimates. Numerical examples are also presented to test thetheoretical analysis. The main results are as followsμFirstly, we use the optimize-then-discreize technique to discretize the second order ellipticoptimal problems and obtain a finite dimensional optimal system. The existence and uniquenessof the system are proved. A priori error estimates are derived. A numerical example is listed toillustrate the theoretical results.Secondly, the semi-discrete finite volume element optimal system of the parabolic optimalproblem is got. We verify the existence and uniqueness of the discrete system and get some apriori error estimates. Back Euler finite volume element scheme and Crank-Nicolson finite volumeelement scheme are used to approximate the semi-discrete system. Convergent analysis of the fully-discrete solution is carried out and some convergent orders are achieved. Numerical experimentsare provided to test all the theoretical results at the end of these contents.Thirdly, we obtain the semi-discrete optimal system of the second order hyperbolic optimalproblems. The existence and uniqueness of the discrete system is proved. Some a priori errorestimates are got for the solution of the semi-discrete optimal system. Two fully-discrete schemesare proposed to approximate the semi-discrete system. One is a finite volume element scheme witha parameter. The other one is Crank-Nicolson finite volume element scheme. Convergent analysisof the Crank-Nicolson finite volume element approximation is carried out and optimal convergent order is obtained. Some numerical experiments are presented to test the theoretical results.