Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T18:00:06.447Z Has data issue: false hasContentIssue false
  • English
  • Français

Solving the Cahn-Hilliard variational inequalitywith 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. [email protected]
Martin Butz
Affiliation:
Universität Regensburg, NWF I-Mathematik, 93040 Regensburg, Germany. [email protected]
Harald Garcke
Affiliation:
Universität Regensburg, NWF I-Mathematik, 93040 Regensburg, Germany. [email protected]
Get access

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-constraintoptimizationsaddle 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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

R.A. Adams, Sobolev spaces, Pure and Applied Mathematics 65. Academic Press, New York-London (1975).
Banas, L. and Nürnberg, R., A multigrid method for the Cahn-Hilliard equation with obstacle potential. Appl. Math. Comput. 213 (2009) 290–303.
Barrett, J.W., Blowey, J.F. and Garcke, H., Finite element approximation of the Cahn–Hilliard equation with degenerate mobility. SIAM J. Numer. Anal. 37 (1999) 286–318. CrossRef
Barrett, J.W., Nürnberg, R. and Styles, V., Finite element approximation of a void electromigration model. SIAM J. Numer. Anal. 42 (2004) 738–772. CrossRef
L. Blank, H. Garcke, L. Sarbu and V. Styles, Primal-dual active set methods for Allen-Cahn variational inequalities with non-local constraints. Preprint SPP1253-09-01 (2009).
Blowey, J.F. and Elliott, C.M., The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy. I. Mathematical analysis. Eur. J. Appl. Math. 2 (1991) 233–280. CrossRef
Blowey, J.F. and Elliott, C.M., The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy. II. Numerical analysis. Eur. J. Appl. Math. 3 (1992) 147–179. CrossRef
J.F. Blowey and C.M. Elliott, Curvature dependent phase boundary motion and parabolic double obstacle problems, in Degenerate Diffusions, W.-M. Ni, L.A. Peletier and J.L. Vazquez Eds., IMA Vol. Math. Appl. 47, Springer, New York (1993) 19–60.
J.F. Blowey and C.M. Elliott, A phase field model with a double obstacle potential, in Motion by mean curvature, G. Buttazzo and A. Visintin Eds., de Gruyter (1994) 1–22.
Cahn, J.W. and Hilliard, J.E., Free energy of a nonuniform system. I. Interfacial energy. J. Chem. Phys. 28 (1958) 258–267. CrossRef
Capuzzo Dolcetta, I., Vita, S.F. and March, R., Area-preserving curve-shortening flows: From phase separation to image processing. Interfaces and Free Boundaries 4 (2002) 325–434. CrossRef
Chen, X., Global asymptotic limit of solutions of the Cahn-Hilliard equation. J. Differential Geom. 44 (1996) 262–311. CrossRef
Chen, X., Nashed, Z. and Smoothing, L. Qi methods and semismooth methods for nondifferentiable operator equations. SIAM J. Numer. Anal. 38 (2000) 1200–1216. CrossRef
Copetti, M. and Elliott, C.M., Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy. Numer. Math. 63 (1992) 39–65. CrossRef
Davis, T.A., Algorithm 832: UMFPACK, an unsymmetric-pattern multifrontal method. ACM Trans. Math. Soft. 30 (2003) 196–199. CrossRef
Davis, T.A., A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Soft. 34 (2003) 165–195.
Davis, T.A. and Duff, I.S., An unsymmetric-pattern multifrontal method for sparse LU factorization. SIAM J. Matrix Anal. Appl. 18 (1997) 140–158. CrossRef
Duff, I.S. and Reid, J.K., The multifrontal solution of indefinite sparse symmetric linear. ACM Trans. Math. Soft. 9 (1983) 302–325. CrossRef
C.M. Elliott, The Cahn-Hilliard model for the kinetics of phase separation, in Mathematical models for phase change problems, Internat. Ser. Numer. Math. 88, Birkhäuser, Basel (1989).
C.M. Elliott and A.R. Gardiner, One dimensional phase field computations, Numerical Analysis 1993, Proceedings of Dundee Conference, D.F. Griffiths and G.A. Watson Eds., Longman Scientific and Technical (1994) 56–74.
C.M. Elliott and S. Luckhaus, A generalised diffusion equation for phase separation of a multi-component mixture with interfacial free energy. SFB 256, University of Bonn, Preprint 195 (1991).
C.M. Elliott and J. Ockendon, Weak and Variational Methods for Moving Boundary Problems, Pitman Research Notes in Mathematics 59. Pitman (1982).
L.C. Evans, Partial differential equations, Graduate Studies in Mathematics 19. American Mathematical Society, Providence (1998).
A. Friedman, Variational principles and free-boundary problems – Pure and Applied Mathematics. John Wiley & Sons, Inc., New York (1982).
H. Garcke, Mechanical effects in the Cahn-Hilliard model: A review on mathematical results, in Mathematical Methods and Models in phase transitions, A. Miranvielle Ed., Nova Science Publ. (2005) 43–77.
C. Gräser, Analysis und Approximation der Cahn-Hilliard Gleichung mit Hindernispotential. Diplomarbeit, Freie Universität Berlin, Fachbereich Mathematik und Informatik (2004).
C. Gräser and R. Kornhuber, On preconditioned Uzawa-type iterations for a saddle point problem with inequality constraints, in Domain decomposition methods in science and engineering XVI, Lect. Notes Comput. Sci. Eng. 55, Springer, Berlin (2007) 91–102.
Gräser, C. and Kornhuber, R., Nonsmooth Newton methods for set-valued saddle point problems. SIAM J. Numer. Anal. 47 (2009) 1251–1273. CrossRef
Gräser, C. and Kornhuber, R., Multigrid methods for obstacle problems. J. Comput. Math. 27 (2009) 1–44.
Hintermüller, M., Ito, K. and Kunisch, K., The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13 (2002) 865–888. CrossRef
Ito, K. and Kunisch, K., Semi-smooth Newton methods for variational inequalities of the first kind. ESAIM: M2AN 37 (2003) 41–62. CrossRef
Irons, B.M., A frontal solution scheme for finite element analysis. Int. J. Numer. Methods Eng. 2 (1970) 5–32. CrossRef
D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics 88. Academic Press, Inc., New York-London (1980).
Kuhl, E. and Schmid, D.W., Computational modeling of mineral unmixing and growth: An application of the Cahn-Hilliard equation. Comp. Mech. 39 (2007) 439–451. CrossRef
Lions, P.-L. and Mercier, B., Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16 (1979) 964–979. CrossRef
Liu, J.W.H., The multifrontal method for sparse matrix solution: Theory and practice. SIAM Rev. 34 (1992) 82–109. CrossRef
Lowengrub, J. and Truskinovsky, L., Quasi-incompressible Cahn-Hilliard fluids and topological transitions. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454 (1978) 2617–2654. CrossRef
Novick-Cohen, A., The Cahn-Hilliard equation: mathematical and modeling perspectives. Adv. Math. Sci. Appl. 8 (1998) 965–985.
R.L. Pego, Front migration in the nonlinear Cahn–Hilliard equation. Proc. Roy. Soc. London, Ser. A 422 (1989) 116–133.
A. Schmidt and K.G. Siebert, Design of adaptive finite element software: The finite element toolbox ALBERTA, Lect. Notes Comput. Sci. Eng. 42. Springer, Berlin (2005).
Stoth, B., Convergence of the Cahn-Hilliard equation to the Mullins-Sekerka problem in spherical symmetry. J. Diff. Equ. 125 (1996) 154–183. CrossRef
Tremaine, S., On the origin of irregular structure in Saturn's rings. Ast. J. 125 (2003) 894–901. CrossRef
F. Tröltzsch, Optimale Steuerung partieller Differentialgleichungen: Theorie, Verfahren und Anwendungen. Vieweg Verlag (2005).
Zhou, S. and Wang, M.Y., Multimaterial structural topology optimization with a generalized Cahn-Hilliard model of multiphase transition. Struct. Multidisc. Optim. 33 (2007) 89–111. CrossRef