A Posteriori Error Estimates For Mixed Finite Element Solutions Of Quadratic Nonlinear Optimal Control Problems

Optimal control problems are playing an important role in scientific computations and engineering applications,and much attention has been paid to its numerical methods. Recently the finite element method seems to be the most widely used method in computing optimal control problems,and the relevant literature is huge.In this work,we investigate the posteriori error estimates for the mixed finite element approximation of quadratic optimal control problem governed by nonlinear elliptic equation.The state and co-state are approximated by the k-th Raviart-Thomas,Brezzi-Douglas-Marini and Brezzi-Douglas-Fortin-Marini mixed finite element spaces and the control is approximated by piecewise constant functions.Our results are based on the approximation for both the state variables and the control variable.We derive a error estimates for some intermediate errors for the RT,the BDM and the BDFM mixed method,our analysis relied on a decomposition of the flux functions in the spirit of a generalized Hemholtz decomposition,then we obtain a posteriori error estimates for the class of optimal control problems.We are then able to derive a posteriori error estimates,which can guide the mesh refinement process in the adaptive analysis.We consider the most useful type of constraints,a posteriori error estimators are established for the constrained optimal control problems.In our experiments,the optimization problem were solved numerically by a preconditioned projection algorithm,we use different meshes for the approximation of the state and the control,and use the posteriori error estimators as the mesh refinement indicator.Then it is clear that the adaptive finite element method is more efficient.