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.
Keywords: Cahn-Hilliard equation, active-set methods, semi-smooth Newton methods, gradient flows, PDE-constraint optimization, saddle point structure
@article{COCV_2011__17_4_931_0,
author = {Blank, Luise and Butz, Martin and Garcke, Harald},
title = {Solving the {Cahn-Hilliard} variational inequality with a semi-smooth {Newton} method},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {931--954},
year = {2011},
publisher = {EDP Sciences},
volume = {17},
number = {4},
doi = {10.1051/cocv/2010032},
mrnumber = {2859859},
zbl = {1233.35132},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2010032/}
}
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 SP - 931 EP - 954 VL - 17 IS - 4 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2010032/ DO - 10.1051/cocv/2010032 LA - en ID - COCV_2011__17_4_931_0 ER -
%0 Journal Article %A Blank, Luise %A Butz, Martin %A Garcke, Harald %T Solving the Cahn-Hilliard variational inequality with a semi-smooth Newton method %J ESAIM: Control, Optimisation and Calculus of Variations %D 2011 %P 931-954 %V 17 %N 4 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2010032/ %R 10.1051/cocv/2010032 %G en %F COCV_2011__17_4_931_0
Blank, Luise; Butz, Martin; Garcke, Harald. Solving the Cahn-Hilliard variational inequality with a semi-smooth Newton method. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 4, pp. 931-954. doi: 10.1051/cocv/2010032
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