American Institute of Mathematical Sciences

Advanced
November  2010, 27(4): 1511-1533. doi: 10.3934/dcds.2010.27.1511

A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions

Laurence Cherfils 1, , Madalina Petcu 2, and  Morgan Pierre 2,

1. 

Université de La Rochelle, Laboratoire de Mathématiques Images et Applications EA 3165, Avenue Michel Crépeau, 17042 La Rochelle Cedex 1, France
2. 

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

Received  December 2009 Revised  February 2010 Published  March 2010

We consider a finite element space semi-discretization of the Cahn-Hilliard equation with dynamic boundary conditions. We prove optimal error estimates in energy norms and weaker norms, assuming enough regularity on the solution. When the solution is less regular, we prove a convergence result in some weak topology. We also prove the stability of a fully discrete problem based on the backward Euler scheme for the time discretization. Some numerical results show the applicability of the method.
Keywords: backward Euler scheme, semilinear parabolic problem, Finite element method, error estimates, Łojasiewicz inequality., linearized Crank-Nicolson scheme.
Mathematics Subject Classification: 65M60, 65M1.
Citation: Laurence Cherfils, Madalina Petcu, Morgan Pierre. A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1511-1533. doi: 10.3934/dcds.2010.27.1511
[1]

Yingwen Guo, Yinnian He. Fully discrete finite element method based on second-order Crank-Nicolson/Adams-Bashforth scheme for the equations of motion of Oldroyd fluids of order one. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2583-2609. doi: 10.3934/dcdsb.2015.20.2583
[2]

Dongho Kim, Eun-Jae Park. Adaptive Crank-Nicolson methods with dynamic finite-element spaces for parabolic problems. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 873-886. doi: 10.3934/dcdsb.2008.10.873
[3]

Sondre Tesdal Galtung. A convergent Crank-Nicolson Galerkin scheme for the Benjamin-Ono equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1243-1268. doi: 10.3934/dcds.2018051
[4]

Weidong Zhao, Jinlei Wang, Shige Peng. Error estimates of the $\theta$-scheme for backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 905-924. doi: 10.3934/dcdsb.2009.12.905
[5]

Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641
[6]

Benoît Merlet, Morgan Pierre. Convergence to equilibrium for the backward Euler scheme and applications. Communications on Pure & Applied Analysis, 2010, 9 (3) : 685-702. doi: 10.3934/cpaa.2010.9.685
[7]

Xiaojie Wang. Weak error estimates of the exponential Euler scheme for semi-linear SPDEs without Malliavin calculus. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 481-497. doi: 10.3934/dcds.2016.36.481
[8]

Alexander Zlotnik. The Numerov-Crank-Nicolson scheme on a non-uniform mesh for the time-dependent Schrödinger equation on the half-axis. Kinetic & Related Models, 2015, 8 (3) : 587-613. doi: 10.3934/krm.2015.8.587
[9]

Nora Aïssiouene, Marie-Odile Bristeau, Edwige Godlewski, Jacques Sainte-Marie. A combined finite volume - finite element scheme for a dispersive shallow water system. Networks & Heterogeneous Media, 2016, 11 (1) : 1-27. doi: 10.3934/nhm.2016.11.1
[10]

François Alouges. A new finite element scheme for Landau-Lifchitz equations. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 187-196. doi: 10.3934/dcdss.2008.1.187
[11]

Jie Shen, Xiaofeng Yang. Error estimates for finite element approximations of consistent splitting schemes for incompressible flows. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 663-676. doi: 10.3934/dcdsb.2007.8.663
[12]

Yanqing Wang. A semidiscrete Galerkin scheme for backward stochastic parabolic differential equations. Mathematical Control & Related Fields, 2016, 6 (3) : 489-515. doi: 10.3934/mcrf.2016013
[13]

Konstantinos Chrysafinos, Efthimios N. Karatzas. Symmetric error estimates for discontinuous Galerkin approximations for an optimal control problem associated to semilinear parabolic PDE's. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1473-1506. doi: 10.3934/dcdsb.2012.17.1473
[14]

Hawraa Alsayed, Hussein Fakih, Alain Miranville, Ali Wehbe. Finite difference scheme for 2D parabolic problem modelling electrostatic Micro-Electromechanical Systems. Electronic Research Announcements, 2019, 26: 54-71. doi: 10.3934/era.2019.26.005
[15]

Maurizio Grasselli, Morgan Pierre. Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2393-2416. doi: 10.3934/cpaa.2012.11.2393
[16]

Monika Eisenmann, Etienne Emmrich, Volker Mehrmann. Convergence of the backward Euler scheme for the operator-valued Riccati differential equation with semi-definite data. Evolution Equations & Control Theory, 2019, 8 (2) : 315-342. doi: 10.3934/eect.2019017
[17]

Dominik Hafemeyer, Florian Mannel, Ira Neitzel, Boris Vexler. Finite element error estimates for one-dimensional elliptic optimal control by BV-functions. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019041
[18]

Nobuyuki Kato, Norio Kikuchi. Campanato-type boundary estimates for Rothe's scheme to parabolic partial differential systems with constant coefficients. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 737-760. doi: 10.3934/dcds.2007.19.737
[19]

Janosch Rieger. The Euler scheme for state constrained ordinary differential inclusions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2729-2744. doi: 10.3934/dcdsb.2016070
[20]

Alain Haraux. Some applications of the Łojasiewicz gradient inequality. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2417-2427. doi: 10.3934/cpaa.2012.11.2417

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (47)
  • HTML views (0)
  • Cited by (17)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences