Location of Repository

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

Downloaded from
http://etheses.dur.ac.uk/475/1/Thesis.pdf

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

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.