A new algorithm, called symmetric inertial alternating direction method of multipliers (SIADMM), is designed for separable convex optimization problems with linear constraints in this paper. The convergence rate of the SIADMM is proved to be $\mathcal{O}(1/ \sqrt{k})$. Two kinds of elliptic equation constrained optimization problems, the unconstrained cases as well as the box-constrained cases of the distributed control and the Robin boundary control, are analyzed theoretically and solved numerically. First, the existence and uniqueness of the solutions to these problems are proved. Second, these continuous optimization problems are transformed into discrete optimization problems by the finite element method, and then the discrete optimization problems are solved by the proposed SIADMM. Numerical experiments with different problems are investigated to demonstrate the efficiency of the SIADMM. And the numerical performance of the SIADMM is better than the performance of the ADMM. Moreover, the numerical results show that the convergence rate of the SIADMM tends to be faster than $\mathcal{O}(1/ \sqrt{k})$ in calculation process.