Adaptive Spectral Element Method For Elliptic Partial Differential Equations And Their Optimal Control Problems

With partial differential equation for the equation of state of the optimal control problem in the modern science and chemical,biological,environmental and other engineering field has a widespread application,such as air pollution control problem,electrochemical control problem in the current density and the number of bacteria in biological engineering control and so on can be summed up in partial differential equation of the optimal control problem.It is often difficult to obtain the exact solutions of these control problems,so efficient numerical solutions are very important for the wide application of optimal control problems.At present,finite element method,adaptive finite element method and spectral method have been successfully applied to the numerical calculation of some optimal control problems,and spectral method has unique advantages for solving fully smooth optimal control problems.It is not known whether spectral element method and its adaptive can be widely used in optimal control problems as adaptive finite element method.Therefore,further study on the spectral element method and its adaptive calculation of optimal control problems has certain academic significance and application value.In this paper,the hN adaptive spectral element method for elliptic partial differential equation and its optimal control problem is studied systematically,and its adaptive algorithms are discussed and analyzed respectively.In chapter 1,we introduce the application background of PDES optimal control problems,analyze the advantages and disadvantages of several classical numerical methods for solving these control problems,and expound the achievements and progress in solving PDES optimal control problems.In chapter 2,the hN adaptive spectral element method of elliptic partial differential equation model is studied.The spectral element function space is constructed and the hN spectral element discrete scheme of elliptic partial differential equation is derived.The prior error estimation and posterior error estimation of numerical solutions are proved theoretically,and the hN adaptive spectral element algorithm is designed.The correctness of the theoretical results is verified by a large number of numerical experiments,and the high efficiency of the adaptive algorithm for solving elliptic partial differential equations is demonstrated.In chapter 3,we mainly study hN adaptive spectral element method for control constrained elliptic optimal control problems.The hN spectral element discrete scheme of the optimal control problem is constructed,the prior error estimation of the solution of the control problem is theoretically proved,the posterior error estimator is constructed,and the hN adaptive algorithm is designed.A posteriori error estimator is used as the encryption standard to guide the local encryption of the grid so that the nodes in the region with poor function regularity are distributed more densely.Finally,the gradient projection algorithm is used to solve the discrete system,and the correctness of the theoretical results is verified by a large number of numerical experiments.It is shown that the adaptive spectral element method has many advantages in solving this kind of optimal control problems.