Error Estimates for Finite Element Approximations of Subgradients of Objective Functions in Optimal Control of the Obstacle Problem

We derive and present error estimates for a FEM-based approximation of a particular Clarke subgradient for the reduced objective function arising in optimal control of the obstacle problem. This Clarke subgradient for the reduced objective function can be computed by using an adjoint state that solves a Dirichlet problem on the complement of the strictly active set. Using finite element solutions of the obstacle problem, we construct discrete and convergent inner and outer approximations of this set. To show that our approximations are suitable and convergent, a detailed study of the topological structure of the strictly active set under appropriate assumptions is necessary. Based on the inner approximation, we solve the Dirichlet problem and obtain an upper bound for the error using the outer approximation. This upper bound converges to zero. We present numerical examples to test our estimates.