A finite element method for elliptic optimal control problem in the unbounded domain \({\mathbb {R}}^2\) is proposed in this paper. Initially, the domain truncation technique is employed to tackle the optimality conditions for the problem. Based on the property of the solution in a sufficiently large domain, truncation error estimates are obtained. Subsequently, a priori and a posteriori error estimates of the problem are strictly derived from the error estimation results of the finite element method. Finally, numerical examples are provided to validate the effectiveness and rationality of the theoretical results.
