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.

Appeared in

  • Non-Smooth and Complementarity-Based Distributed Parameter Systems: Simulation and Hierarchical Optimization, M. Hintermüller, R. Herzog, Ch. Kanzow, M. Ulbrich, S. Ulbrich, eds., vol. 172 of International Series of Numerical Mathematics, Birkhäuser, Springer Nature Switzerland AG, Cham, 2022, pp. 211--240.

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