Improved error estimates for optimal control of the Stokes problem with pointwise tracking in three dimensions

Abstract

. This work is motivated by recent interest in the topic of pointwise tracking type optimal control problems for the Stokes problem. Pointwise tracking consists of point evaluations in the objective functional which lead to Dirac measures appearing as source terms of the adjoint problem. Consider- ing bounds for the control allows for improved regularity results for the exact solution and improved approximation error estimates of its numerical coun- terpart. We show a sub-optimal convergence result in three dimensions that nonetheless improves the results known from the literature. Finally, we offer supporting numerical experiments and insights towards optimal approximation error estimates.