Hostname: page-component-848d4c4894-xfwgj Total loading time: 0 Render date: 2024-07-02T11:31:20.822Z Has data issue: false hasContentIssue false
  • English
  • Français

A posteriori estimates for the Cahn–Hilliard equation with obstacle free energy

Published online by Cambridge University Press:  12 June 2009

Ľubomír Baňas  and
Robert Nürnberg
Ľubomír Baňas
Affiliation:
Department of Mathematics and the Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK. L.Banas@hw.ac.uk
Robert Nürnberg
Affiliation:
Department of Mathematics, Imperial College London, London, SW7 2AZ, UK.
Get access

Abstract

We derive a posteriori estimates for a discretization in space of the standard Cahn–Hilliard equation with a double obstacle free energy. The derived estimates are robust and efficient, and in practice are combined with a heuristic time step adaptation. We present numerical experiments in two and three space dimensions and compare our method with an existing heuristic spatial mesh adaptation algorithm.

Keywords

Cahn–Hilliard equationobstacle free energylinear finite elementsa posteriori estimatesadaptive numerical methods
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 5 , September 2009 , pp. 1003 - 1026
Copyright
© EDP Sciences, SMAI, 2009

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

Alikakos, N.D., Bates, P.W. and Chen, X.F., The convergence of solutions of the Cahn–Hilliard equation to the solution of the Hele–Shaw model. Arch. Rational Mech. Anal. 128 (1994) 165–205. CrossRef
Ľ. Baňas, R. Nürnberg, Adaptive finite element methods for Cahn–Hilliard equations. J. Comput. Appl. Math. 218 (2008) 2–11. CrossRef
Ľ. Baňas, R. Nürnberg, Finite element approximation of a three dimensional phase field model for void electromigration. J. Sci. Comp. 37 (2008) 202–232. CrossRef
Ľ. Baňas and R. Nürnberg, Phase field computations for surface diffusion and void electromigration in ${\mathbb R}^3$ . Comput. Vis. Sci. (2008), doi: 10.1007/s00791-008-0114-0.
Barrett, J.W. and Blowey, J.F., Finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy. Numer. Math. 77 (1997) 1–34. CrossRef
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 phase field model for void electromigration. SIAM J. Numer. Anal. 42 (2004) 738–772. CrossRef
Blowey, J.F. and Elliott, C.M., The Cahn–Hilliard gradient theory for phase separation with non-smooth free energy. Part I: Mathematical analysis. European J. Appl. Math. 2 (1991) 233–279. CrossRef
Blowey, J.F. and Elliott, C.M., The Cahn–Hilliard gradient theory for phase separation with non-smooth free energy. Part II: Numerical analysis. European J. Appl. Math. 3 (1992) 147–179. CrossRef
Braess, D., A posteriori error estimators for obstacle problems – another look. Numer. Math. 101 (2005) 415–421. CrossRef
Cahn, J.W., On spinodal decomposition. Acta Metall. 9 (1961) 795–801. CrossRef
Cahn, J.W. and Hilliard, J.E., Free energy of a non-uniform system. I. Interfacial free energy. J. Chem. Phys. 28 (1958) 258–267. CrossRef
Chen, X., Spectrum for the Allen–Cahn, Cahn–Hilliard, and phase-field equations for generic interfaces. Comm. Partial Differ. Equ. 19 (1994) 1371–1395. CrossRef
Chen, Z. and Nochetto, R.H., Residual type a posteriori error estimates for elliptic obstacle problems. Numer. Math. 84 (2000) 527–548. CrossRef
Clément, P., Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9 (1975) 77–84.
Elliott, C.M. and Songmu, Z., On the Cahn–Hilliard equation. Arch. Rational Mech. Anal. 96 (1986) 339–357. CrossRef
Elliott, C.M., French, D.A. and Milner, F.A., A second order splitting method for the Cahn–Hilliard equation. Numer. Math. 54 (1989) 575–590. CrossRef
Feng, X. and Prohl, A., Error analysis of a mixed finite element method for the Cahn–Hilliard equation. Numer. Math. 99 (2004) 47–84. CrossRef
Feng, X. and Wu, H., A posteriori error estimates for finite element approximations of the Cahn–Hilliard equation and the Hele–Shaw flow. J. Comput. Math. 26 (2008) 767–796.
Hintermüller, M. and Hoppe, R.H.W., Goal-oriented adaptivity in control constrained optimal control of partial differential equations. SIAM J. Control Optim. 47 (2008) 1721–1743. CrossRef
Hintermüller, M., Hoppe, R.H.W., Iliash, Y. and Kieweg, M., An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints. ESAIM: COCV 14 (2008) 540–560. CrossRef
Kim, J., Kang, K. and Lowengrub, J., Conservative multigrid methods for Cahn–Hilliard fluids. J. Comput. Phys. 193 (2004) 511–543. CrossRef
Modica, L., Gradient theory of phase transitions with boundary contact energy. Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987) 487–512. CrossRef
Moon, K.-S., Nochetto, R.H., von Petersdorff, T. and Zhang, C.-S., A posteriori error analysis for parabolic variational inequalities. ESAIM: M2AN 41 (2007) 485–511. CrossRef
Nochetto, R.H. and Wahlbin, L.B., Positivity preserving finite element approximation. Math. Comp. 71 (2002) 1405–1419. CrossRef
Pego, R.L., Front migration in the nonlinear Cahn-Hilliard equation. Proc. Roy. Soc. London Ser. A 422 (1989) 261–278. CrossRef
Veeser, A., Efficient and reliable a posteriori error estimators for elliptic obstacle problems. SIAM J. Numer. Anal. 39 (2001) 146–167. CrossRef
A. Veeser, On a posteriori error estimation for constant obstacle problems, in Numerical methods for viscosity solutions and applications (Heraklion, 1999), M. Falcone and C. Makridakis Eds., Ser. Adv. Math. Appl. Sci. 59, World Sci. Publ., River Edge, USA (2001) 221–234.
R. Verfürth, A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Teubner-Wiley, New York (1996).