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

Niklas Behringer

doi:10.3934/mcrf.2020038

Article Contents

2021, Volume 11, Issue 2: 313-328. Doi: 10.3934/mcrf.2020038

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

Niklas Behringer^,

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

^* Corresponding author: Niklas Behringer
^* Corresponding author: Niklas Behringer

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

Received: February 29, 2020

Revised: June 30, 2020

Early access: August 2020

Published: June 2021

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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. 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.

Keywords:

Linear-quadratic optimal control problem,
Stokes equation,
Dirac measures,
a priori error estimates,
maximum-norm estimates.

Mathematics Subject Classification: Primary:35Q35, 49M25;Secondary:35R06, 65N15, 65N30.

Citation:

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Figure 1. Threshold visualization of the first component of a solution $ \vec{q}_h $ to Problem (16)

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Figure 2. 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

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