Nonsmooth Schur–Newton methods for multicomponent Cahn–Hilliard systems

We present globally convergent nonsmooth Schur–Newton methods for the solution of discrete multicomponent Cahn–Hilliard systems with logarithmic and obstacle potentials. The method solves the nonlinear set-valued saddle-point problems arising from discretization by implicit Euler methods in time and first-order finite elements in space without regularization. Efficiency and robustness of the convergence speed for vanishing temperature is illustrated by numerical experiments.

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

[2]  I. Steinbach,et al.  A phase field concept for multiphase systems , 1996 .

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

[4]  Junseok Kim,et al.  A numerical method for the ternary Cahn--Hilliard system with a degenerate mobility , 2009 .

[5]  John W. Barrett,et al.  An optimal error bound for a finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy , 1999 .

[6]  John W. Barrett,et al.  An Error Bound for the Finite Element Approximation of a Model for Phase Separation of a Multi-Compo , 1996 .

[7]  Robert Nürnberg,et al.  The degenerate and non-degenerate deep quench obstacle problem: A numerical comparison , 2013, Networks Heterog. Media.

[8]  Harald Garcke,et al.  Solving the Cahn-Hilliard variational inequality with a semi-smooth Newton method , 2011 .

[9]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[10]  Carsten Gräser.  Globalization of Nonsmooth Newton Methods for Optimal Control Problems , 2008 .

[11]  Moulay Hicham Tber,et al.  An adaptive finite-element Moreau–Yosida-based solver for a non-smooth Cahn–Hilliard problem , 2011, Optim. Methods Softw..

[12]  Ralf Kornhuber,et al.  Time discretizations of anisotropic Allen–Cahn equations , 2013 .

[13]  D. de Fontaine,et al.  An analysis of clustering and ordering in multicomponent solid solutions—I. Stability criteria , 1972 .

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

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

[16]  Ralf Kornhuber,et al.  On constrained Newton linearization and multigrid for variational inequalities , 2002, Numerische Mathematik.

[17]  Harald Garcke,et al.  A multi-phase Mullins–Sekerka system: matched asymptotic expansions and an implicit time discretisation for the geometric evolution problem , 1998, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[18]  Charles M. Elliott,et al.  Numerical analysis of a model for phase separation of a multi- component alloy , 1996 .

[19]  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.

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

[21]  Ralf Kornhuber,et al.  On multigrid methods for vector-valued Allen-Cahn equations , 2003 .

[22]  Maya Neytcheva,et al.  Efficient numerical solution of discrete multi-component Cahn-Hilliard systems , 2014, Comput. Math. Appl..

[23]  John W. Barrett,et al.  An Improved Error Bound for a Finite Element Approximation of a Model for Phase Separation of a Mult , 1999 .

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

[25]  John W. Cahn,et al.  Spinodal decomposition in ternary systems , 1971 .

[26]  Michael Hinze,et al.  A Globalized Semi-smooth Newton Method for Variational Discretization of Control Constrained Elliptic Optimal Control Problems , 2012, Constrained Optimization and Optimal Control for Partial Differential Equations.

[27]  Carsten Gräser,et al.  Convex minimization and phase field models , 2011 .

[28]  Harald Garcke,et al.  Nonlocal Allen–Cahn systems: analysis and a primal–dual active set method , 2013 .

[29]  John W. Barrett,et al.  An improved error bound for a finite element approximation of a model for phase separation of a multi-component alloy with a concentration dependent mobility matrix , 2001, Numerische Mathematik.

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

[31]  Oliver Sander,et al.  The dune-subgrid module and some applications , 2009, Computing.

[32]  Ralf Kornhuber,et al.  Robust Multigrid Methods for Vector-valued Allen–Cahn Equations with Logarithmic Free Energy , 2006 .

[33]  Oliver Sander,et al.  Truncated Nonsmooth Newton Multigrid Methods for Convex Minimization Problems , 2009 .

[34]  Gene H. Golub,et al.  A Note on Preconditioning for Indefinite Linear Systems , 1999, SIAM J. Sci. Comput..