Solving the Cahn-Hilliard variational inequality with a semi-smooth Newton method

Luise Blank; Martin Butz; Harald Garcke

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 4, page 931-954
  • ISSN: 1292-8119

Abstract

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The Cahn-Hilliard variational inequality is a non-standard parabolic variational inequality of fourth order for which straightforward numerical approaches cannot be applied. We propose a primal-dual active set method which can be interpreted as a semi-smooth Newton method as solution technique for the discretized Cahn-Hilliard variational inequality. A (semi-)implicit Euler discretization is used in time and a piecewise linear finite element discretization of splitting type is used in space leading to a discrete variational inequality of saddle point type in each time step. In each iteration of the primal-dual active set method a linearized system resulting from the discretization of two coupled elliptic equations which are defined on different sets has to be solved. We show local convergence of the primal-dual active set method and demonstrate its efficiency with several numerical simulations.

How to cite

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Blank, Luise, Butz, Martin, and Garcke, Harald. "Solving the Cahn-Hilliard variational inequality with a semi-smooth Newton method." ESAIM: Control, Optimisation and Calculus of Variations 17.4 (2011): 931-954. <http://eudml.org/doc/221916>.

@article{Blank2011,
abstract = { The Cahn-Hilliard variational inequality is a non-standard parabolic variational inequality of fourth order for which straightforward numerical approaches cannot be applied. We propose a primal-dual active set method which can be interpreted as a semi-smooth Newton method as solution technique for the discretized Cahn-Hilliard variational inequality. A (semi-)implicit Euler discretization is used in time and a piecewise linear finite element discretization of splitting type is used in space leading to a discrete variational inequality of saddle point type in each time step. In each iteration of the primal-dual active set method a linearized system resulting from the discretization of two coupled elliptic equations which are defined on different sets has to be solved. We show local convergence of the primal-dual active set method and demonstrate its efficiency with several numerical simulations. },
author = {Blank, Luise, Butz, Martin, Garcke, Harald},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Cahn-Hilliard equation; active-set methods; semi-smooth Newton methods; gradient flows; PDE-constraint optimization; saddle point structure; primal-dual active set method; discrete variational inequality},
language = {eng},
month = {11},
number = {4},
pages = {931-954},
publisher = {EDP Sciences},
title = {Solving the Cahn-Hilliard variational inequality with a semi-smooth Newton method},
url = {http://eudml.org/doc/221916},
volume = {17},
year = {2011},
}

TY - JOUR
AU - Blank, Luise
AU - Butz, Martin
AU - Garcke, Harald
TI - Solving the Cahn-Hilliard variational inequality with a semi-smooth Newton method
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2011/11//
PB - EDP Sciences
VL - 17
IS - 4
SP - 931
EP - 954
AB - The Cahn-Hilliard variational inequality is a non-standard parabolic variational inequality of fourth order for which straightforward numerical approaches cannot be applied. We propose a primal-dual active set method which can be interpreted as a semi-smooth Newton method as solution technique for the discretized Cahn-Hilliard variational inequality. A (semi-)implicit Euler discretization is used in time and a piecewise linear finite element discretization of splitting type is used in space leading to a discrete variational inequality of saddle point type in each time step. In each iteration of the primal-dual active set method a linearized system resulting from the discretization of two coupled elliptic equations which are defined on different sets has to be solved. We show local convergence of the primal-dual active set method and demonstrate its efficiency with several numerical simulations.
LA - eng
KW - Cahn-Hilliard equation; active-set methods; semi-smooth Newton methods; gradient flows; PDE-constraint optimization; saddle point structure; primal-dual active set method; discrete variational inequality
UR - http://eudml.org/doc/221916
ER -

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