  In this paper, we carry out the numerical analysis of a nonsmooth quasilinear
elliptic optimal control problem, where the coefficient in the divergence term
of the corresponding state equation is not differentiable with respect to the
state variable. Despite the lack of differentiability of the nonlinearity in
the quasilinear elliptic equation, the corresponding control-to-state operator
is of class $C^1$ but not of class $C^2$. Analogously, the discrete
control-to-state operators associated with the approximated control problems
are proven to be of class $C^1$ only. By using an explicit second-order
sufficient optimality condition, we prove a priori error estimates for a
variational approximation, a piecewise constant approximation, and a continuous
piecewise linear approximation of the continuous optimal control problem. The
numerical tests confirm these error estimates.
