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

Luise Blank; Martin Butz; Harald Garcke

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 4, page 931-954
  • ISSN: 1292-8119

Abstract

top
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.

How to cite

top

Blank, Luise, Butz, Martin, and Garcke, Harald. "Solving the Cahn-Hilliard variational inequality with a semi-smooth Newton method." ESAIM: Control, Optimisation and Calculus of Variations 17.4 (2011): 931-954. <http://eudml.org/doc/272776>.

@article{Blank2011,
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.},
author = {Blank, Luise, Butz, Martin, Garcke, Harald},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Cahn-Hilliard equation; active-set methods; semi-smooth Newton methods; gradient flows; PDE-constraint optimization; saddle point structure; primal-dual active set method; discrete variational inequality},
language = {eng},
number = {4},
pages = {931-954},
publisher = {EDP-Sciences},
title = {Solving the Cahn-Hilliard variational inequality with a semi-smooth Newton method},
url = {http://eudml.org/doc/272776},
volume = {17},
year = {2011},
}

TY - JOUR
AU - Blank, Luise
AU - Butz, Martin
AU - Garcke, Harald
TI - Solving the Cahn-Hilliard variational inequality with a semi-smooth Newton method
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2011
PB - EDP-Sciences
VL - 17
IS - 4
SP - 931
EP - 954
AB - 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.
LA - eng
KW - Cahn-Hilliard equation; active-set methods; semi-smooth Newton methods; gradient flows; PDE-constraint optimization; saddle point structure; primal-dual active set method; discrete variational inequality
UR - http://eudml.org/doc/272776
ER -

References

top
  1. [1] R.A. Adams, Sobolev spaces, Pure and Applied Mathematics 65. Academic Press, New York-London (1975). Zbl0314.46030MR450957
  2. [2] L. Banas and R. Nürnberg, A multigrid method for the Cahn-Hilliard equation with obstacle potential. Appl. Math. Comput.213 (2009) 290–303. Zbl1168.65386MR2536652
  3. [3] J.W. Barrett, J.F. Blowey and H. Garcke, Finite element approximation of the Cahn–Hilliard equation with degenerate mobility. SIAM J. Numer. Anal.37 (1999) 286–318. Zbl0947.65109MR1742748
  4. [4] J.W. Barrett, R. Nürnberg and V. Styles, Finite element approximation of a void electromigration model. SIAM J. Numer. Anal.42 (2004) 738–772. Zbl1076.78012MR2084234
  5. [5] 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). Zbl1272.65060MR3039798
  6. [6] J.F. Blowey and C.M. Elliott, The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy. I. Mathematical analysis. Eur. J. Appl. Math. 2 (1991) 233–280. Zbl0797.35172MR1123143
  7. [7] J.F. Blowey and C.M. Elliott, The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy. II. Numerical analysis. Eur. J. Appl. Math. 3 (1992) 147–179. Zbl0810.35158MR1166255
  8. [8] 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. Zbl0794.35092MR1246337
  9. [9] 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. Zbl0809.35168MR1277388
  10. [10] J.W. Cahn and J.E. Hilliard, Free energy of a nonuniform system. I. Interfacial energy. J. Chem. Phys. 28 (1958) 258–267. 
  11. [11] I. Capuzzo Dolcetta, S.F. Vita and R. March, Area-preserving curve-shortening flows: From phase separation to image processing. Interfaces and Free Boundaries4 (2002) 325–434. Zbl1021.35129MR1935642
  12. [12] X. Chen, Global asymptotic limit of solutions of the Cahn-Hilliard equation. J. Differential Geom.44 (1996) 262–311. Zbl0874.35045MR1425577
  13. [13] X. Chen, Z. Nashed and L. Qi, Smoothing methods and semismooth methods for nondifferentiable operator equations. SIAM J. Numer. Anal.38 (2000) 1200–1216. Zbl0979.65046MR1786137
  14. [14] M. Copetti and C.M. Elliott, Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy. Numer. Math.63 (1992) 39–65. Zbl0762.65074MR1182511
  15. [15] T.A. Davis, Algorithm 832: UMFPACK, an unsymmetric-pattern multifrontal method. ACM Trans. Math. Soft.30 (2003) 196–199. Zbl1072.65037MR2075981
  16. [16] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Soft.34 (2003) 165–195. Zbl1072.65036
  17. [17] T.A. Davis and I.S. Duff, An unsymmetric-pattern multifrontal method for sparse LU factorization. SIAM J. Matrix Anal. Appl.18 (1997) 140–158. Zbl0884.65021MR1428205
  18. [18] I.S. Duff and J.K. Reid, The multifrontal solution of indefinite sparse symmetric linear. ACM Trans. Math. Soft.9 (1983) 302–325. Zbl0515.65022MR791968
  19. [19] 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). Zbl0692.73003MR1038064
  20. [20] 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. Zbl0802.65123MR1267755
  21. [21] 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). 
  22. [22] C.M. Elliott and J. Ockendon, Weak and Variational Methods for Moving Boundary Problems, Pitman Research Notes in Mathematics 59. Pitman (1982). Zbl0476.35080MR650455
  23. [23] L.C. Evans, Partial differential equations, Graduate Studies in Mathematics 19. American Mathematical Society, Providence (1998). Zbl0902.35002MR1625845
  24. [24] A. Friedman, Variational principles and free-boundary problems – Pure and Applied Mathematics. John Wiley & Sons, Inc., New York (1982). Zbl0671.49001MR679313
  25. [25] 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. MR2590944
  26. [26] C. Gräser, Analysis und Approximation der Cahn-Hilliard Gleichung mit Hindernispotential. Diplomarbeit, Freie Universität Berlin, Fachbereich Mathematik und Informatik (2004). 
  27. [27] 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. MR2334094
  28. [28] C. Gräser and R. Kornhuber, Nonsmooth Newton methods for set-valued saddle point problems. SIAM J. Numer. Anal.47 (2009) 1251–1273. Zbl1190.49035MR2485452
  29. [29] C. Gräser and R. Kornhuber, Multigrid methods for obstacle problems. J. Comput. Math.27 (2009) 1–44. Zbl1199.65401MR2493556
  30. [30] M. Hintermüller, K. Ito and K. Kunisch, The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim.13 (2002) 865–888. Zbl1080.90074MR1972219
  31. [31] K. Ito and K. Kunisch, Semi-smooth Newton methods for variational inequalities of the first kind. ESAIM: M2AN 37 (2003) 41–62. Zbl1027.49007MR1972649
  32. [32] B.M. Irons, A frontal solution scheme for finite element analysis. Int. J. Numer. Methods Eng.2 (1970) 5–32. Zbl0252.73050
  33. [33] 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). Zbl0457.35001MR567696
  34. [34] E. Kuhl and D.W. Schmid, Computational modeling of mineral unmixing and growth: An application of the Cahn-Hilliard equation. Comp. Mech.39 (2007) 439–451. Zbl1162.74004
  35. [35] P.-L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal.16 (1979) 964–979. Zbl0426.65050MR551319
  36. [36] J.W.H. Liu, The multifrontal method for sparse matrix solution: Theory and practice. SIAM Rev.34 (1992) 82–109. Zbl0919.65019MR1156290
  37. [37] J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci.454 (1978) 2617–2654. Zbl0927.76007MR1650795
  38. [38] A. Novick-Cohen, The Cahn-Hilliard equation: mathematical and modeling perspectives. Adv. Math. Sci. Appl.8 (1998) 965–985. Zbl0917.35044MR1657208
  39. [39] R.L. Pego, Front migration in the nonlinear Cahn–Hilliard equation. Proc. Roy. Soc. London, Ser. A 422 (1989) 116–133. Zbl0701.35159MR997638
  40. [40] 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). Zbl1068.65138MR2127659
  41. [41] B. Stoth, Convergence of the Cahn-Hilliard equation to the Mullins-Sekerka problem in spherical symmetry. J. Diff. Equ.125 (1996) 154–183. Zbl0851.35011MR1376064
  42. [42] S. Tremaine, On the origin of irregular structure in Saturn's rings. Ast. J.125 (2003) 894–901. 
  43. [43] F. Tröltzsch, Optimale Steuerung partieller Differentialgleichungen: Theorie, Verfahren und Anwendungen. Vieweg Verlag (2005). Zbl1142.49001
  44. [44] S. Zhou and M.Y. Wang, Multimaterial structural topology optimization with a generalized Cahn-Hilliard model of multiphase transition. Struct. Multidisc. Optim.33 (2007) 89–111. Zbl1245.74077MR2291576

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.