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A numerical analysis of the Cahn–Hilliard equation with non-permeable walls

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Abstract

In this article we consider the numerical analysis of the Cahn–Hilliard equation in a bounded domain with non-permeable walls, endowed with dynamic-type boundary conditions. The dynamic-type boundary conditions that we consider here have been recently proposed in Ruiz Goldstein et al. (Phys D 240(8):754–766, 2011) in order to describe the interactions of a binary material with the wall. The equation is semi-discretized using a finite element method for the space variables and error estimates between the exact and the approximate solution are obtained. We also prove the stability of a fully discrete scheme based on the backward Euler scheme for the time discretization. Numerical simulations sustaining the theoretical results are presented.

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Acknowledgments

The authors would like to thank Professor Alain Miranville for suggesting this problem, as well as for fruitful discussions.

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Authors and Affiliations

  1. Laboratoire de Mathématiques Images et Applications EA 3165, Université de La Rochelle, Avenue Michel Crépeau, 17042 , La Rochelle Cedex 1, France

    Laurence Cherfils

  2. Laboratoire de Mathématiques et Applications UMR CNRS 6086, Université de Poitiers, BP 30179, Téléport 2, Boulevard Marie et Pierre Curie, 86962 , Futuroscope Chasseneuil, France

    Madalina Petcu

  3. The Institute of Mathematics of the Romanian Academy, Bucharest, Romania

    Madalina Petcu

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  1. Laurence Cherfils
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  2. Madalina Petcu
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Correspondence to Madalina Petcu.

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Cherfils, L., Petcu, M. A numerical analysis of the Cahn–Hilliard equation with non-permeable walls. Numer. Math. 128, 517–549 (2014). https://doi.org/10.1007/s00211-014-0618-0

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