  An accelerated block coordinate descent (ABCD) method in Hilbert space is
analyzed to solve the sparse optimal control problem via its dual. The finite
element approximation of this method is investigated and convergence results
are presents. Based on the second order growth condition of the dual objective
function, we show that iteration sequence of dual variables has the iteration
complexity of $O(1/k)$. Moreover, we also prove iteration complexity for the
primal problem. Two types of mesh-independence for ABCD method are proved,
which asserts that asymptotically the infinite dimensional ABCD method and
finite dimensional discretizations have the same convergence property, and the
iterations of ABCD method remain nearly constant as the discretization is
refined.
