In this paper, we investigate a variational discretization approximation of elliptic optimal control problems with control constraints. The state and the co-state are approximated by piecewise linear functions, while the control is not directly discretized. By using some proper intermediate variables, we derive a second-order convergence in <TEX>$L^2$</TEX>-norm and superconvergence between the numerical solution and elliptic projection of the exact solution in <TEX>$H^1$</TEX>-norm or the gradient of the exact solution and recovery gradient in <TEX>$L^2$</TEX>-norm. Then we construct a posteriori error estimates by using the superconvergence results and do some numerical experiments to confirm our theoretical results.