# Error estimates for the numerical approximation of a non-smooth quasilinear elliptic control problem

In this paper, we carry out the numerical analysis of a non-smooth quasilinear elliptic optimal control problem, where the coefficient in the divergence term of the corresponding state equation is a finitely PC^2 (continuous and C^2 apart from finitely many points) function in the state variable. Although the nonlinearity of the quasilinear elliptic equation is non-smooth, 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. An explicit formula of a second-order generalized derivative of the cost functional is also established. We then exploit a second-order sufficient optimality condition to prove a priori error estimates for a variational and a piecewise constant approximation of the continuous optimal control problem.

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