Spectral Element Method For Several Types Of Optimal Control Problems Governed By Elliptic Partial Differential Equations

Optimal control problem constrained by partial differential equation-s(PDE control problem for short)is widely and important in engineer-ing,economy,biotechnology and other fields.Therefore,the numerical methods of PDE control problem have always been a hot research field.In this paper,the spectral element methods for solving optimal control problems governed by elliptic partial differential equations are system-atically studied.The spectral element approximation for solving several different types of elliptic control problems with control constraints,state constraints and control-state constraints are discussed respectively.The method not only has the characteristics of regional flexibility of the fi-nite element method,but also has the advantages of high accuracy of the spectral method.The research results show that the spectral element method can effectively solve these kinds of elliptic control problems,the numerical solution has high accuracy,and the accuracy of numerical so-lution can be improved by increasing the order of polynomial or reducing the size of subdomains.Control constraint is a typical case of PDE control problems.In this paper,the spectral element approximation of control constrained elliptic control problems is studied,in which the control variable is constrained by L~2-norm.Firstly,using the theory of optimality condition derivation,the optimality condition of the problem is deduced,and the property of multiplierλis discussed.Then,a spectral element discrete scheme for L~2-norm constrained elliptic control problem and its optimality condition are established.Then,the a prior error estimation of spectral element approximation is deduced with the help of auxiliary system,and the H~1-L~2and L~2-L~2a posterior error estimates results of spectral element approximation are obtained respectively.Finally,the gradient projection algorithm is used to solve the problem.The results show that the spectral element solution can obtain high accuracy,and the numerical results are consistent with the theoretical results.Another typical case of PDE control problem is state constraint.Therefore,the spectral element solution of state constrained elliptic con-trol problem is further studied,in which the state variable is constrained by integral condition.The spectral element discrete scheme of the prob-lem is constructed,and the optimality conditions of the continuous prob-lem and the discrete problem are derived respectively.By proving the boundedness of the numerical solutions and introducing the auxiliary sys-tem,the a priori error estimates of spectral element approximation for variables such as control,state and conjugate state are derived respec-tively,and the H~1-L~2a posteriori error and L~2-L~2a posteriori error analysis of spectral element approximation are established.Numerical experiments show that the accuracy of spectral element approximation can be improved by increasing the polynomial order N or reducing the size h of the subdomains.Compared with the elliptic control problems with integral state con-straint,the spectral element analysis of the elliptic control problem with H~1-norm constraint is more complex.Firstly,the optimality conditions of the state H~1-norm constrained elliptic control problems are strictly derived,and the related properties of multiplierλis proved.Then,by introducing an appropriate spectral element space,the spectral element discrete scheme of the problem and its optimality conditions are estab-lished.Then,the auxiliary system is introduced and the a prior error analysis of spectral element approximation are derived by using the meth-ods of classification discussion,and the H~1-L~2and L~2-L~2a posterior error estimates results of spectral element approximation are obtained through the auxiliary system.Finally,numerical experiments are car-ried out to verify the theoretical results,and the results show that the spectral element method can effectively solve the elliptic control problem with H~1-norm constraint on state.Another kind of case for PDE control problems is that the con-trol and state are constrained at the same time,which is more complex than the control problems with pure control constraint or pure state con-straint.In this paper,the a posteriori error estimates of spectral element approximation for control-state constrained elliptic control problem is s-tudied,in which the control variables and state variables are constrained by integration.Firstly,the optimality conditions of the control-state in-tegral constrained elliptic control problem are derived,and the related properties of the multiplierλare proved.Then,the spectral element approximation scheme of the problem and its optimality conditions is established,and the H~1-L~2and L~2-L~2a posterior error estimates of spectral element approximation are obtained with the help of the auxiliary system and the properties of the related interpolation operator.Finally,the effectiveness of the method is verified by numerical experiments.