Convergence And Quasi-optimality Of Adaptive Finite Element Methods For Nonlinear Elliptic Optimal Control Problems

To solve the partial differential optimal control problem,the infinite dimensional optimal problems need to be transformed into a finite dimensional optimal problem where it is usually realized by finite element methods.There are a few aspects should be considered in the selection of finite element discretization scheme.The first is to solve the optimization problem.The scale of the optimal problems depend on the number of degrees of freedom of the finite element meshes which is hoped that the number of degrees of freedom can be reduced as much as possible.The second is the approximation accuracy problem.The nonsmooth coefficients of constrained partial differential optimal control problem will produce nonsmooth solutions,which will reduce the computational accuracy.The adaptive finite element methods can effectively take into account the above two issues,and can reduce the optimization scale and improve the calculation accuracy.The main idea of the adaptive finite element methods is to measure the approximation error which is calculated through the error indicators on each dissection element by using the computable amount of finite element solutions and given data.The elements with larger error estimators are selected for marking and encrypting to form a new mesh.Then the degree of freedom is distributed in the area where the solutions have singularities.This paper will investigate the convergence and quasi-optimality of adaptive finite element methods for nonlinear elliptic optimal control problems.The main contents include four parts.In the first part,a class of nonlinear elliptic optimal control problems with quadratic functional is studied.By using Lagrange multiplier methods,the optimal conditions which are coupled by state equations,dual equations and variational inequalities are obtained.The control variables are discretized by the piecewise constant finite element method,the state variables and the adjoint state variables are discretized by the piecewise linear finite element method,and the variational inequalities are treated by the variational discretization method.In the second part,the a posteriori error estimates of the above nonlinear elliptic optimal control problems is studied.The error equations are dealt with by the nonlinear linearization methods proposed by Professor Miliner.The residual a posterior error estimates of the finite element solutions are obtained by using the idea of bubble and new bubble functions,the dual arguments,the interpolation operators and the projection operators.In the third part,the convergence of the adaptive finite element method for the above nonlinear elliptic optimal control problems is studied.The local perturbation property,the reduction property on error estimator and oscillation,and the quasi-orthogonality are obtained by combining the nonlinear term linearization method,the D?fler property,the dual arguments and the common basic inequalities.The contraction of the adaptive finite element algorithms is proved by using above three properties.In the last part,the quasi-optimality of the adaptive finite element method for the above nonlinear elliptic optimal control problem is studied.A class of function approximation is introduced to obtain the discrete local upper bound and quasi orthogonality which are obtained by using the nonlinear term linearization methods,dual arguments,common basic inequalities,interpolation operators and projection operators.The quasi-optimality of the adaptive finite element algorithm has proved by combining the D?fler property.This paper considered the nonlinear elliptic optimal control problems.The optimal conditions of the nonlinear elliptic boundary value problems has been established.It has obtained the residual type a posteriori error estimates of the finite element solutions of the nonlinear elliptic optimal control problems.The convergence and quasi-optimality of the adaptive finite element method are proved by using the above optimal conditions.The theoretical analysis is verified by numerical examples,and the numerical simulation of a class of nonlinear optimal control problems is to realize.