Convergence and quasi-optimality of an adaptive finite element method for elliptic Robin boundary control problem

In this paper we focus on the convergence and quasi-optimality of an adaptive finite element method for elliptic Robin boundary control problems. We use piecewise linear finite elements to approximate the state and the adjoint state variables, and the variational discretization to approximate the control variable. Under mild assumption on the initial mesh, we prove the contraction property, for the sum of the energy errors of the state and adjoint state and the scaled error estimator, on two consecutive adaptive loops. The resulting linear convergence yields the quasi-optimal convergence rate for the AFEM algorithm applied to our problem. Additionally, some numerical results are provided to support our theoretical analysis.