Research On Algorithm For Two Kind Of PDE-constrained Optimal Control Problems With Interfaces

Optimal control problems have a wide range of applications in engineering and science,such as materials science,temperature control,air pollution,etc.In practical applications,due to the diversity of constituent material materials,there exist interfaces between different substances,so that the mathematical model of this class of problems belongs to the interface problems,and the mathematical model of the control process of this type problem is the PDEconstrained optimal control problem with interface.Due to the poor smoothness and regularity of interface,it is even more difficult to solve the exact solution.Hence,it is very important to study the numerical solution to this class of problem.In this paper,we mainly study the adaptive finite element method for the following two kinds of PDE-constrained optimal control problems with interfaces:In the first part,we study the elliptic optimal control problem with interface.Considering the smoothness and regularity of the interface are low,we develop the adaptive immersed finite element method to solve this problem.Firstly,the control variable is discretized by piecewise constant functions.The state and the adjoint state are discretized by continuous piecewise linear functions.We get the a posteriori error estimation.Secondly,in order to save the computational work and improve the convergence rate of the control,we combine the variational discretization with the finite element method to discretize the optimality conditions.Finally,we give a lot of numerical examples and the results of numerical experiments are provided to confirm the a posteriori error estimators are effective.In the second part,we consider the parabolic optimal control problem with the interface.Firstly,the problem is discretized by the variational discretization.We use the linear finite element for the approximation of the space variables and use the backward Euler method for the time variable.Secondly,we derive the a posteriori error estimates of the variational discretization for the class of problem.