Finite Volume Element Method For Elliptic Partial Differential Equation Distribution And Neumann Boundary Control Constraints

In the past few decades,the optimal control mathematical theo-ry has rapidly developed into an important and independent applied mathematical field,which is widely used in physical engineering,bi-ological engineering,social science and other fields.In these fields,there are many interesting problems,in which we need to solve the minimization problem for a given cost functional under the differen-tial equation and some constraints.Numerical method is needed to solve the problem.The finite volume element method(FVM)has at-tracted more and more attention because of its local conservation of physical quantities.In this paper,the finite volume element method with elliptic partial differential equation distribution and boundary control constraints on rectangular regions is studied.The optimal control problem studied in this paper is constrained by the elliptic partial differential equation,which contains distribut-ed control and Neumann boundary control.For this kind of opti-mal control problem with dual control variables,the optimal system of the control problem is first obtained by using Lagrange multipli-er method.Secondly,the finite volume element method is used to discrete the optimal system.Based on the nonlinear coupling vari-ational form,the trial function space is selected as the linear finite element space,while the test function space is selected as the simple piecewise constant function space.Thus,the finite volume element method is constructed for the dual control problem under the PDE constraint,and the results of the second-order precision of the state,adjoint state,distribution control and Neumann boundary control are obtained.Finally,numerical experiments are given to verify the effectiveness of the proposed method.