International Journal of Difference Equations (IJDE). ISSN 0973-6069, Volume 15, Number 1, (2020). 11-29 c Research India Publications https://dx.doi.org/10.37622/IJDE/15.1.2020.11-29 Finite Element and Split Bregman Methods for Solving a Family of Optimal Control Problem with Partial Differential Equation Constraint Mahmoud Lotﬁ Department of Applied Mathematics, University of Kurdistan, Sanandaj, Iran. Abstract In this article we will discuss the solution of elliptic optimal control problem. First, by using the ﬁnite element method we obtain 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, and hence it saves time and memory requirement. Then we use the split Bregman method for solving this problem, and examples show the speed and accuracy of split Bregman methods for solving this type of problems . We also use the SQP method for solving the examples and compare with split Bregman method. Keywords: Split Bregman Method, Optimal Control with Elliptic Partial Differential Equation Constraint, Finite Element Method. 1. INTRODUCTION 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 paper we will discuss the