Optimization and Engineering

, Volume 19, Issue 3, pp 629–662 | Cite as

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

  • Michael Hintermüller
  • Michael Hinze
  • Christian KahleEmail author
  • Tobias Keil
Research Article


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.


Optimal control Two phase flow Diffuse interfaces Goal-oriented adaptivity Dual-weighted residuals MPEC 


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Michael Hintermüller
    • 1
    • 2
  • Michael Hinze
    • 3
  • Christian Kahle
    • 4
    Email author
  • Tobias Keil
    • 2
  1. 1.Weierstrass InstituteBerlinGermany
  2. 2.Department MathematikHumboldt Universität zu BerlinBerlinGermany
  3. 3.Fachbereich MathematikUniversität HamburgHamburgGermany
  4. 4.Zentrum Mathematik, M17Technische Universität MünchenGarchingGermany

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