Geodesic
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

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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.

DOI : 10.1051/cocv/2010032
Classification : 35K55, 35K85, 90C33, 49N90, 80A22, 82C26, 65M60
Keywords: Cahn-Hilliard equation, active-set methods, semi-smooth Newton methods, gradient flows, PDE-constraint optimization, saddle point structure 
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     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},
     publisher = {EDP-Sciences},
     volume = {17},
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     year = {2011},
     doi = {10.1051/cocv/2010032},
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     zbl = {1233.35132},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2010032/}
}
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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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