In recent decades optimal control problems with partial differential equation constraints have been studied extensively. These issues are very complex and the numerical solution of such problems is of great importance. In this article we will discuss the solution of elliptic optimal control problem. First, by using the finite element method we obtain to gain the discrete form of the problem. The obtained discrete problem is actually a large scale constrained optimization problem. Solving this optimization problem with traditional methods is difficult and requires a lot of CPU time and memory. But split Bergman method converts the constrained problem to an unconstrained problem, and hence it saves time and memory requirement. We then use the split Bregman iterative methods for solving this problem, and examples show the speed and accuracy of split Bregman iterative methods for solving this type of problems. We also use the SQP method for solving the problem and compare with split Bregman method.

Keywords: Finite Element Method, Split Bregman Method

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