Numerical analysis of a nonsmooth quasilinear elliptic control problem

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 non-linearity in the quasilinear elliptic equation, the corresponding control-to-state operator is of class C1 but not of class C2. Analogously, the discrete control-to-state operators associated with the approximated control problems are proven to be of class C1 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.