# 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

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topBlank, 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/272776>.

@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},

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/272776},

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

PY - 2011

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/272776

ER -

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