# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2020038

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

 Department of Mathematics, Technical University of Munich, Boltzmannstrasse 3, 85748 Garching, Germany

* Corresponding author: Niklas Behringer

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project number 188264188/GRK1754.

Received  March 2020 Revised  July 2020 Published  August 2020

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. Considering bounds for the control allows for improved regularity results for the exact solution and improved approximation error estimates of its numerical counterpart. 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.

Citation: Niklas Behringer. Improved error estimates for optimal control of the Stokes problem with pointwise tracking in three dimensions. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2020038
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##### References:
Threshold visualization of the first component of a solution $\vec{q}_h$ to Problem (16)
Error $||\bar q_{n}-\bar q_h||_{L^2(\Omega)}$ for cellwise constant control discretization and different choices for the bounds $\vec a$ and $\vec b$. $\bar q_{n}$ denotes the approximate solution on a finer mesh