Researches On Low Order Nonconforming Finite Element Methods For Optimal Control Problems

In this dissertation, we focus on the nonconforming finite element and mixed finite element approximations to the optimal control problems governed by partial differential equations, which include the elliptic type, Stokes type, parabolic type, and so on. Optimal control problems have been widely employed in many engineering fields, such as atmospheric pollution control, temperature control, oil exploiting, image processing, and so on. Most of the optimal control problems always have large scale computations, which need to be solved quickly, so the high accuracy numerical methods become the first priority to study. In fact, the nonconforming elements have obvious advantages in the parallel computations comparing with the conforming ones. Because the exact solutions of the optimal control problems always have low order regularities, so low order elements will be chosen to approximate the variables in the first place. By using some special properties of the low order nonconforming elements, such as the interpolation operator equaling to the Rieze projection, the consistency error being one order higher than the interpolation error and etc., the optimal order error estimates or the superclose results are derived via some novel techniques and methods. By employing the post-processing technique, the global superconvergence properties are then obtained based on the superclose results. In addition, the last two chapters discuss the nonconforming and characteristic-nonconforming finite element methods for the interface problems and the convection-dominated transport problems, respectively. At the same time, some numerical results are provided to verify the theoretical analysis. This research provides new choices for numerical computations of the optimal control problems and has important theoretical significance and practical applications to expanding the nonconforming finite element methods.The thesis consists of eight chapters: Chapter 1 the preface. The research status and historical background of the optimal control problems are introduced in this chapter.Some basic knowledge and frequently used symbols are provided. At last, the structure arrangement of this thesis is presented as well.In Chapter 2, a high accuracy nonconforming EQrot1element method for elliptic optimal control problem is introduced. And the superclose properties for the state, co-state and the control variables are obtained. The superconvergence results are obtained for the state and co-state variables in a broken energy norm by the post-processing technique. At the same time, some numerical examples are provided to verify the theoretical analysis.Lastly, the results are extended to some other famous nonconforming elements cases.The related conclusion about this part has been published in Applied Mathematics and Computation.In Chapter 3, another elliptic optimal control problem is analyzed. The co-state variable appeared in the objective function, so we constructed a mixed finite element scheme, in which the co-state is approximated by a pair of nonconforming elements, and the state and the control are approximated by the piecewise constant element. The scheme just meets the necessary LBB conditions. The optimal order error estimates are obtained by some special properties of the elements. Finally, some numerical experiments are provided to verify the feasibility of the scheme. The related work about this part has been published in Computers and Mathematics with Applications.In chapter 4, a mixed finite element method is proposed for the Stokes optimal control problem by using the nonconforming EQrot1+P0 finite element pair. The scheme also meets the LBB condition. The optimal order error estimates are derived.In chapter 5, a nonconforming finite element method is proposed for a nonsmooth elliptic optimal control problem. Firstly, the nonconforming finite element analysis is given for the nonsmooth elliptic equations. The optimal order error estimates and the global superconvergence results are obtained. The solution of such nonsmooth optimal control problem may not be unique in many cases, so it is not meaningful to estimate the errors between the numerical solution and the exact solution. Therefore, the goal-oriented error estimate is considered, indicating that the finite element solution tended to an exact solution in the sense of goal function. The related results have been submitted to Acta Mathematicae Applicatae Sinica.In chapter 6, the nonconforming finite element methods are proposed for the parabolic optimal control problems. The time discretization is based on difference methods. The control variable is approximated by the piecewise constant elements, for which the superclose properties and the negative norm error estimates are derived. The state and co-state are approximated by the nonconforming EQrot1elements, the global superclose and superconvergence results are obtained in the broken energy norm by using the post-processing technique.In chapter 7, the elliptic and parabolic interface problems are approximated by the lowest order P1 triangular finite element methods. Given the fact that the finite element space only contains the first order polynomial, the difficulty of error estimate near the interface is conquered, and the optimal order error estimates are derived. Some numerical experiments are given to verify the theoretical analysis. The related results of this chapter are to be published by Applied Mathematics and Mechanics.In chapter 8, a characteristic-nonconforming finite element method is proposed for solving a convection-dominated transport problem. Using a novel error estimation technique, the global superclose and superconvergence results are derived in a broken energy norm, which improves the results of References [86, 87]. And the numerical results coincide with the theoretical analysis. The related work of this part has been published in Mathematical Methods in the Applied Sciences.