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.

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