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

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.

[1]  Stefano Finzi Vita,et al.  Area-preserving curve-shortening flows: from phase separation to image processing , 2002 .

[2]  Kazufumi Ito,et al.  Semi–Smooth Newton Methods for Variational Inequalities of the First Kind , 2003 .

[3]  Ralf Kornhuber,et al.  Multigrid Methods for Obstacle Problems , 2008 .

[4]  Timothy A. Davis,et al.  A column pre-ordering strategy for the unsymmetric-pattern multifrontal method , 2004, TOMS.

[5]  Charles M. Elliott,et al.  The Cahn–Hilliard gradient theory for phase separation with non-smooth free energy Part II: Numerical analysis , 1991, European Journal of Applied Mathematics.

[6]  P. Lions,et al.  Splitting Algorithms for the Sum of Two Nonlinear Operators , 1979 .

[7]  Harald Garcke,et al.  Finite Element Approximation of the Cahn-Hilliard Equation with Degenerate Mobility , 1999, SIAM J. Numer. Anal..

[8]  S. Tremaine,et al.  On the Origin of Irregular Structure in Saturn's Rings , 2002, astro-ph/0211149.

[9]  Charles M. Elliott,et al.  The Cahn-Hilliard Model for the Kinetics of Phase Separation , 1989 .

[10]  Ralf Kornhuber,et al.  On preconditioned Uzawa-type iterations for a saddle point problem with inequality constraints , 2007 .

[11]  John W. Barrett,et al.  Finite Element Approximation of a Phase Field Model for Void Electromigration , 2004, SIAM J. Numer. Anal..

[12]  Bruce M. Irons,et al.  A frontal solution program for finite element analysis , 1970 .

[13]  Charles M. Elliott,et al.  Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy , 1992 .

[14]  Joseph W. H. Liu,et al.  The Multifrontal Method for Sparse Matrix Solution: Theory and Practice , 1992, SIAM Rev..

[15]  Ralf Kornhuber,et al.  Nonsmooth Newton Methods for Set-Valued Saddle Point Problems , 2009, SIAM J. Numer. Anal..

[16]  J. E. Hilliard,et al.  Free Energy of a Nonuniform System. I. Interfacial Free Energy , 1958 .

[17]  Robert Nürnberg,et al.  A multigrid method for the Cahn-Hilliard equation with obstacle potential , 2009, Appl. Math. Comput..

[18]  D. Kinderlehrer,et al.  An introduction to variational inequalities and their applications , 1980 .

[19]  Charles M. Elliott,et al.  `A generalised diffusion equation for phase separation of a multi-component mixture with interfacial free energy' , 1991 .

[20]  Shiwei Zhou,et al.  Multimaterial structural topology optimization with a generalized Cahn–Hilliard model of multiphase transition , 2006 .

[21]  Kunibert G. Siebert,et al.  Design of Adaptive Finite Element Software - The Finite Element Toolbox ALBERTA , 2005, Lecture Notes in Computational Science and Engineering.

[22]  Timothy A. Davis,et al.  Algorithm 832: UMFPACK V4.3---an unsymmetric-pattern multifrontal method , 2004, TOMS.

[23]  Gene H. Golub,et al.  Numerical solution of saddle point problems , 2005, Acta Numerica.

[24]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[25]  F. Tröltzsch.  Optimale Steuerung partieller Differentialgleichungen , 2005 .

[26]  J. Lowengrub,et al.  Quasi–incompressible Cahn–Hilliard fluids and topological transitions , 1998, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[27]  Timothy A. Davis,et al.  An Unsymmetric-pattern Multifrontal Method for Sparse Lu Factorization , 1993 .

[28]  Xiaojun Chen,et al.  Smoothing Methods and Semismooth Methods for Nondifferentiable Operator Equations , 2000, SIAM J. Numer. Anal..

[29]  Kazufumi Ito,et al.  The Primal-Dual Active Set Strategy as a Semismooth Newton Method , 2002, SIAM J. Optim..

[30]  Lavinia Sarbu,et al.  Primal‐dual active set methods for Allen–Cahn variational inequalities with nonlocal constraints , 2010 .

[31]  E. Kuhl,et al.  Computational modeling of mineral unmixing and growth An application of the Cahn – Hilliard equation , 2006 .

[32]  Harald Garcke,et al.  Mathematik Primal-dual active set methods for Allen-Cahn variational inequalities with non-local constraints , 2009 .

[33]  Harald Garcke,et al.  Mechanical Effects in the Cahn-Hilliard Model: A Review on Mathematical Results , 2005 .

[34]  G. Burton.  Sobolev Spaces , 2013 .

[35]  C. M. Elliott,et al.  Weak and variational methods for moving boundary problems , 1982 .

[36]  A. Friedman.  Variational principles and free-boundary problems , 1982 .

[37]  John K. Reid,et al.  The Multifrontal Solution of Indefinite Sparse Symmetric Linear , 1983, TOMS.

[38]  Xinfu Chen,et al.  Global asymptotic limit of solutions of the Cahn-Hilliard equation , 1996 .

[39]  Barbara Stoth,et al.  Convergence of the Cahn-Hilliard Equation to the Mullins-Sekerka Problem in Spherical Symmetry , 1996 .

[40]  J. F. Bloweyl,et al.  A phase-field model with a double obstacle potential , 1994 .

[41]  James F. Blowey,et al.  Curvature Dependent Phase Boundary Motion and Parabolic Double Obstacle Problems , 1993 .

[42]  Robert L. Pego,et al.  Front migration in the nonlinear Cahn-Hilliard equation , 1989, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.