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

Published online by Cambridge University Press:  18 August 2010

Luise Blank ,
Martin Butz  and
Harald Garcke
Luise Blank
Affiliation:
Universität Regensburg, NWF I-Mathematik, 93040 Regensburg, Germany. luise.blank@mathematik.uni-regensburg.de
Martin Butz
Affiliation:
Universität Regensburg, NWF I-Mathematik, 93040 Regensburg, Germany. luise.blank@mathematik.uni-regensburg.de
Harald Garcke
Affiliation:
Universität Regensburg, NWF I-Mathematik, 93040 Regensburg, Germany. luise.blank@mathematik.uni-regensburg.de
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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.

Keywords

Cahn-Hilliard equationactive-set methodssemi-smooth Newton methodsgradient flowsPDE-constraint optimizationsaddle point structure
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 4 , October 2011 , pp. 931 - 954
Copyright
© EDP Sciences, SMAI, 2010

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