A majorized accelerated block coordinate descent (mABCD) method in Hilbert space is analyzed to solve the sparse optimal control problem via its dual. The finite element approximation method is investigated and convergence results are presented. The attractive $O(1/k^2)$ iteration complexity of ABCD method for the dual objective function values can be achieved. However the convergence for the sequence of the dual variables is not obvious. In order to accomplish this goal, a local second order growth condition for the dual objective function is presented. Then, the iteration complexity of $O(1/k)$ for the iteration sequence of dual variables can be proved. Furthermore, making use of the relationship between the primal problem and dual problem, the iteration complexity results for the primal problem are proved.