RT1Mixed Finite Element Methods Of Optimal Control Problems

Optimal control problems have been widely met in all kinds of practical prob-lems, such as, temperature control problems, electric field and magnetic field con-trol problems, air pollution control problems, number of bacilli control problems inbiology engineering, electrochemical machining design problems, etc. Efcient nu-merical methods are among the keys to successful applications of optimal controlin practical areas. Finite element approximation of optimal control problems playsa very important role in numerical methods for these problems. There have beenextensively studies on this aspect. When the objective functional in the controlproblems contains the gradient of the state variables, mixed finite element meth-ods should be used for discretization of the state equation. In this paper, we willinvestigate a priori, superconvergence and a posteriori error estimates for ellipticand parabolic optimal control problems by RT1mixed finite element methods.The paper consists of three parts. In the first part of this paper, we studythe linear elliptic optimal control problems. We first transform the minimizationproblem into a coupled system of state equation, co-state equation and a vari-ational inequality. The state and the co-state are approximated by RT1mixedfinite element spaces and the control is discretized by piecewise linear functions.For linear elliptic optimal control problems with an integral constraint on control,using the Green’s functions which were proposed by H. Chen and Z. Jiang and theduality argument, we obtain an optimal L∞-error estimates for optimal controlproblems. For the optimal control problems with an obstacle constraint, we deriveL∞-error estimate for the control variable by use of the method of the basis func-tions presented by C. Meyer and A. Ro¨sch. Finally, we present some numericalexamples which confirms our theoretical results.In the second part, we discuss the nonlinear elliptic optimal control problems.The state and the co-state variables are still approximated by RT1mixed finite el-ement spaces, the control variable is discretized by piecewise constant functions orpiecewise linear functions. For nonlinear elliptic optimal control problems with anintegral constraint on control, by using postprocessing technique, we derive a globalsuperconvergent result in L2-norm when the control variable was approximated bypiecewise constant functions. Moreover, we obtain L∞-error estimates for semilin- ear elliptic optimal control problems when the control variable was approximatedby piecewise linear functions. Finally, we present some numerical results.In the last part, we will analyze the optimal control problems governed byparabolic equations. First, we consider the variational discretization which wasfirst proposed by M. Hinze for the constrained optimal control problem, we do notdiscretize the space of admissible control but implicitly utilize the relation betweenco-state and the control for the discretization of the control. We analyze the fullydiscrete mixed finite element scheme and obtain a priori error estimates for thestate, the co-state and the control. We also present some numerical examples.Next, using elliptic construction method (proposed by C. Makridakis and R. H.Nochetto), we get a posteriori error estimates for linear parabolic optimal controlproblems.