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 conditions bring a large amount of computation. Therefore, we study the domain decomposition 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 convergence 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=N1；step3:LetUnh,k+1/2=Unh,kp(Unh,k+G*Pn1h,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 subdomains,reducing operation time and improve efficiency.In addition,the algorithm here is on a general domain and its decompositions are general.
