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Mathematical and Numerical Analysis of a Pair of Coupled Cahn-Hilliard Equations with a Logarithmic Potential

By M ALI AHMED AL-GHAFLI

Abstract

Mathematical and numerical analysis has been undertaken for a pair of coupled Cahn-Hilliard equations with a logarithmic potential and with homogeneous Neumann boundary conditions. This pair of coupled equations arises in a phase separation model of thin film of binary liquid mixture. Global existence and uniqueness of a weak solution to the problem is proved using Faedo-Galerkin method. Higher regularity results of the weak solution are established under further regular requirements on the initial data. Further, continuous dependence on the initial data is presented.\ud \ud Numerically, semi-discrete and fully-discrete piecewise linear finite element approximations to the continuous problem are proposed for which existence, uniqueness and various stability estimates of the approximate solutions are proved. Semi-discrete and fully-discrete error bounds are derived where the time discretisation error is optimal. An iterative method for solving the resulting nonlinear algebraic system is introduced and linear stability analysis in one space dimension is studied. Finally, numerical experiments illustrating some of the theoretical results are performed in one and two space dimensions

Year: 2010
OAI identifier: oai:etheses.dur.ac.uk:475
Provided by: Durham e-Theses

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Citations

  1. (1991). A generalised diffusion equation for phase separation of a multi-component mixture with interfacial free energy.
  2. (2000). A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations. doi
  3. (2005). A reaction-diffusion system of λ−ω type. Part I: Mathematical analysis. doi
  4. (1995). An error bound for the finite element approximation of the Cahn-Hilliard equation with logarithmic free energy. doi
  5. (2001). An improved error bound for a finite element approximation of a model for phase separation of a multi-component alloy with a concentration dependent mobility matrix. doi
  6. (1999). An improved error bound for a finite element approximation of a model for phase separation of a multi-component alloy. doi
  7. (1996). An Introduction to Partial Differential Equations. doi
  8. (1975). Analyse numerique d’un proble´ me de stefan a deux phase par une methode d’elements finis. doi
  9. (2003). Analysis of a reaction diffusion system of λ-ω type. doi
  10. (1977). Cone conditions and properties of Sobolev spaces. doi
  11. (2007). Convergence to equilibrium for the Cahn-Hilliard equation with a logarithmic free energy. Nonlinear Analysis, doi
  12. (1999). Discrete Versions of Gronwall lemma and their application to the numerical analysis of parabolic problems.
  13. (1997). e. Galerkin Finite Element Methods for Parabolic Problems. doi
  14. (2001). Elliptic Partial Differential Equations of Second Order. doi
  15. (1985). Elliptic Problems in Non-Smooth Domains, Pitman Adanced Publishing Program,
  16. (1987). Error analysis of the enthalpy method for the Stefan problem. doi
  17. (1999). Finite element approximation of a fourth order nonlinear degenerate parabolic equation. doi
  18. (1998). Finite element approximation of a model for phase separation of a multi-component alloy with a concentration-dependent mobility matrix. doi
  19. (1999). Finite element approximation of a model for phase separation of a multi-component alloy with nonsmooth free energy and a concentration-dependent mobility matrix. doi
  20. (1997). Finite element approximation of a model for phase separation of a multi-component alloy with nonsmooth-free energy. doi
  21. (2004). Finite element approximation of a nonlinear cross-diffusion population model. doi
  22. (2002). Finite element approximation of an AllenCahn/Cahn-Hilliard system. doi
  23. (2007). Finite element approximation of spatially extended predator-prey interactions with the Holling type II functional response. doi
  24. (1999). Finite element approximation of the CahnHilliard equation with logarithmic free energy. doi
  25. Finite element approximation of transport of reactive solutes in porous media: Part 1. Error estimates for non-equilibrium adsorption process. doi
  26. (1991). Finite element methods for parabolic free boundary problems. doi
  27. (1958). Free energy of a nonuniform system.I. interfacial free energy. doi
  28. (1977). Function Spaces.
  29. (1972). General Lagrange and Hermit interpolation in Rn with applications to finite element methods. Archive for Rational Mechanics and Analysis, doi
  30. (1997). Infinite-Dimensional Dynamical Systems in Mechanics and Physics. doi
  31. (2001). Infinite-Dimensional Dynamical Systems: an Introduction to Dissipative Parabolic PDEs and The Theory of Global Attractors. doi
  32. (1996). Interfacial roughening induced by phase separation. doi
  33. (1984). Navier-Stokes Equations: Theory and Numerical Analysis. doi
  34. (1984). Nonlinear aspects of the Cahn-Hilliard equation. doi
  35. (1990). Nonlinear Elliptic and Evolution Problems and their Finite Element Approximations. doi
  36. (2007). Numerical analysis of a coupled pair of Cahn-Hilliard equations with non-smooth free energy.
  37. (2001). Numerical analysis of a coupled pair of Chan-Hilliard equations.
  38. (1996). Numerical analysis of a model for phase separation of a multi-component alloy. doi
  39. (1992). Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy. doi
  40. (1990). Numerical Solution of Partial Differential Equations by the Finite Element Method. doi
  41. (1990). Numerical studies of phase separation in models of binary alloys and polymer blends. doi
  42. Obstacle Problems doi
  43. (2001). On fully practical finite element approximations of degenerate Cahn-Hilliard systems. doi
  44. (2009). On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions. Communcations On Pure and Applied Analysis, doi
  45. (1969). Quelques M´ ethodes de R´ esolution des Probl´ emes aux Limites non Lin´ eaires. Dunod Gauthier-Villars,
  46. (1975). Sobolev Spaces. doi
  47. (1979). Spinodal decomposition (phase transition via unstable states). doi
  48. (1971). Spinodal decomposition: A reprise. Acta Metall, doi
  49. (1979). Splitting algorithms for the sum of two nonlinear operators. doi
  50. (1988). Study of phase separation dynamics by use of Cell Dynamical system. doi
  51. (1992). The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy. Part II: Numerical analysis, doi
  52. (1989). The Cahn-Hilliard model for the kinetics of phase separation. doi
  53. (1979). The Finite Element Method for Elliptic Problems. doi
  54. (1993). The global dynamics of discrete semilinear parabolic equations. doi
  55. (1987). The gradient theory of phase separations and the minimal interface criterion. Archive for Rational Mechanics and Analysis. doi
  56. (1994). The Mathematical Theory of Finite Element Methods. doi
  57. (1973). The use of numerical integration in finite element methods for solving parabolic doi
  58. (2004). Theory and Practice of Finite Elements. doi
  59. (1977). Theory of dynamic critical phenomena. doi
  60. (1996). Weak and Measure-Valued Solutions to Evolutionary PDEs. doi

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