Error estimates for the numerical approximation of Neumann control problems

We continue the discussion of error estimates for the numerical analysis of Neumann boundary control problems we started in Casas et al. (Comput. Optim. Appl. 31:193-219, 2005). In that paper piecewise constant functions were used to approximate the control and a convergence of order 0(h) was obtained. Here, we use continuous piecewise linear functions to discretize the control and obtain the rates of convergence in L-2(Gamma). Error estimates in the uniform norm are also obtained. We also discuss the approach suggested by Hinze (Comput. Optim. Appl. 30:45-61, 2005) as well as the improvement of the error estimates by making an extra assumption over the set of points corresponding to the active control constraints. Finally, numerical evidence of our estimates is provided.