WIAS Preprint No. 2617, (2019)

Simulation and control of a nonsmooth Cahn--Hilliard Navier--Stokes system with variable fluid densities


  • Gräßle, Carmen
  • Hintermüller, Michael
    ORCID: 0000-0001-9471-2479
  • Hinze, Michael
  • Keil, Tobias

2010 Mathematics Subject Classification

  • 35B65 35J87 35K55 65K15 49K20


  • Two phase flow, POD model order reduction, adaptivity, nonsmooth systems, mathematical programming with equilibrium constraints, optimal control




We are concerned with the simulation and control of a two phase flow model governed by a coupled Cahn--Hilliard Navier--Stokes system involving a nonsmooth energy potential.We establish the existence of optimal solutions and present two distinct approaches to derive suitable stationarity conditions for the bilevel problem, namely C- and strong stationarity. Moreover, we demonstrate the numerical realization of these concepts at the hands of two adaptive solution algorithms relying on a specifically developed goal-oriented error estimator.In addition, we present a model order reduction approach using proper orthogonal decomposition (POD-MOR) in order to replace high-fidelity models by low order surrogates. In particular, we combine POD with space-adapted snapshots and address the challenges which are the consideration of snapshots with different spatial resolutions and the conservation of a solenoidal property.

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