WIAS Preprint No. 2311, (2016)

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



Authors

  • Hintermüller, Michael
    ORCID: 0000-0001-9471-2479
  • Hinze, Michael
  • Kahle, Christian
  • Keil, Tobias

2010 Mathematics Subject Classification

  • 49K20 49M29 65K15 76T10 90C33

Keywords

  • Cahn-Hilliard, C-stationarity, mathemathical programming with equilibrium constraints, Navier-Stokes, non-matched densities, non-smooth potentials, optimal control, adaptive finite element method, goal-oriented error estimation

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 to 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-weighed 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 residual weighted by approximate dual quantities and vice versa as well as various complementary mismatch errors. Details on the numerical realization of the adaptive concept and a report on numerical tests are given.

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