A Domain Decomposition Algorithm And The Posteriori Error Estimates For Optimal Control Problems Governed By Parabolic Equations

In this paper, we study a the optimal control problem governed by linear parabolic equation as below, where T > 0, ydand and there exist two positive numbersα0 andα1 such that 0<α0<α(x)<α1.In the traditional finite element method, three coupled equations in optimality con-ditions bring a large amount of computation. Therefore, we study the domain decom-position algorithm. Extend the domain decomposition algorithm of a single parabolic partial differential equation to control problems, and propose the domain decomposition procedure of the optimal control problem. Let n=1,2,... N,In the fourth part of this paper, we derive a priori error estimates, and gain the con-vergence order:(Δt+h2+h2u+H5/2). In the dinal part,we give the iteration for domain decomposition procedure.Let k be the step of the iteration,given the initial value,then: step1:by the{Unh,k}Nn=1 calculate{Ynh,k+1)Nn=1；step2:by the results of the first step{Ynh,k+1)Nn=1 calculate{Pnh,k+1}0n=N-1；step3:LetUnh,k+1/2=Unh,k-p(Unh,k+G*Pn-1h,k+1),and Unh,k+1=QUnh,k+1/2.we can find Unh,k+1 step4:repeat steps of one to three.Finally,we prove the convergence of the iteration.The advantage of the algorithm is that the area can be decomposed into several small sub-domains,reducing operation time and improve efficiency.In addition,the algorithm here is on a general domain and its decompositions are general.