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

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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Correspondence to Christian Kahle.

Additional information

This research was supported by the German Research Foundation DFG through the SPP 1506, the SPP 1962, and the International Research Training Group IGDK 1754 “Optimization and Numerical Analysis for Partial Differential Equations with Nonsmooth Structures”. It was further supported by the Berlin Mathematical School and by the Research Center MATHEON through project C-SE5 funded by the Einstein Center for Mathematics Berlin.

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Hintermüller, M., Hinze, M., Kahle, C. et al. A goal-oriented dual-weighted adaptive finite element approach for the optimal control of a nonsmooth Cahn–Hilliard–Navier–Stokes system. Optim Eng 19, 629–662 (2018). https://doi.org/10.1007/s11081-018-9393-6

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Keywords

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