Solving the Cahn-Hilliard variational inequality with a semi-smooth Newton method
Universität Regensburg, NWF I-Mathematik, 93040 Regensburg, Germany. firstname.lastname@example.org
Revised: 2 March 2010
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.
Mathematics Subject Classification: 35K55 / 35K85 / 90C33 / 49N90 / 80A22 / 82C26 / 65M60
Key words: Cahn-Hilliard equation / active-set methods / semi-smooth Newton methods / gradient flows / PDE-constraint optimization / saddle point structure
© EDP Sciences, SMAI, 2010