  We consider the variational discretization of a linear-quadratic optimal
control problem with pointwise control and state constraints. In order to allow
for a Fr\'echet smooth norm, the problem is reformulated by means of a
reflexive Sobolev space instead of the space of continuous functions. The
discretization of the state equation yields a family of perturbed optimal
control problems, whose solutions can be computed numerically. We apply an
implicit multifunction theorem (IMT) to the first order necessary conditions to
proof a bound on the perturbation error for these solutions. In order to verify
the abstract regularity condition of the IMT, we compute the Fr\'echet
coderivative of a set-valued representation of the necessary conditions.
Applying our results to an elliptic state equation in two dimensions,
undergoing a simple finite element discretization, we obtain convergence of
order O(h).
