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Optimization and Engineering

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23-06-2018 | Research Article

A goal-oriented dual-weighted adaptive finite element approach for the optimal control of a nonsmooth Cahn–Hilliard–Navier–Stokes system

Authors: Michael Hintermüller, Michael Hinze, Christian Kahle, Tobias Keil

Published in: Optimization and Engineering | Issue 3/2018

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Abstract

This paper is concerned with the development and implementation of an adaptive solution algorithm for the optimal control of a time-discrete Cahn–Hilliard–Navier–Stokes system with variable densities. The free energy density associated with the Cahn–Hilliard system incorporates the double-obstacle potential which yields an optimal control problem for a family of coupled systems in each time instant of a variational inequality of fourth order and the Navier–Stokes equation. A dual-weighted residual approach for goal-oriented adaptive finite elements is presented which is based on the concept of C-stationarity. The overall error representation depends on primal residuals weighted by approximate dual quantities and vice versa as well as various complementarity mismatch errors. Details of the numerical realization of the adaptive concept and a report on the numerical tests are given.

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Title: A goal-oriented dual-weighted adaptive finite element approach for the optimal control of a nonsmooth Cahn–Hilliard–Navier–Stokes system
Authors: Michael Hintermüller
Michael Hinze
Christian Kahle
Tobias Keil
Publication date: 23-06-2018
Publisher: Springer US
Published in: Optimization and Engineering / Issue 3/2018
Print ISSN: 1389-4420
Electronic ISSN: 1573-2924
DOI: https://doi.org/10.1007/s11081-018-9393-6

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