Finite element approximation of an Allen-Cahn/Cahn-Hilliard system

We consider an Allen–Cahn/Cahn–Hilliard system with a non-degenerate mobility and (i) a logarithmic free energy and (ii) a non-smooth free energy (the deep quench limit). This system arises in the modelling of phase separation and ordering in binary alloys. In particular we prove in each case that there exists a unique solution for sufficiently smooth initial data. Further, we prove an error bound for a fully practical piecewise linear finite element approximation of (i) and (ii) in one and two space dimensions (and three space dimensions for constant mobility). The error bound being optimal in the deep quench limit. In addition an iterative scheme for solving the resulting nonlinear discrete system is analysed. Finally some numerical experiments are presented

The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy Part II : numerical analysis.

In this paper we consider the numerical analysis of a parabolic variational inequality arising from a deep quench limit of a model for phase separation in a binary mixture due to Cahn and Hilliard. Stability, convergence and error bounds for a finite element approximation are proven. Numerical simulations in one and two space dimensions are presented

An improved error bound for a finite element approximation of a model for phase separation of a multi-component alloy.

Using the approach of Rulla (1996 SIAM J. Numer. Anal. 33, 68-87) for analysing the time discretization error and assuming more regularity on the initial data, we improve on the error bound derived by Barrett and Blowey (1996 IMA J. Numer. Anal. 16, 257-287) for a fully practical piecewise linear finite element approximation with a backward Euler time discretization of a model for phase separation of a multi-component alloy

A reaction-diffusion system of λ–ω typePart I: Mathematical analysis.

We study two coupled reaction-diffusion equations of the $\lambda$–$\omega$ type [11] in $d\,{\le}\,3$ space dimensions, on a convex bounded domain with a $C^2$ boundary. The equations are close to a supercritical Hopf bifurcation in the reaction kinetics and are model equations for oscillatory reaction-diffusion equations. Global existence, uniqueness and continuous dependence on initial data of strong and weak solutions are proved using the classical Faedo-Galerkin method of Lions [15] and compactness arguments. We also present a complete case study for the application of this method to systems of nonlinear reaction-diffusion equations

Finite element approximation of a model for phase separation of a multi-component alloy with a concentration dependent mobility matrix,

We consider a model for phase separation of a multi-component alloy with a concentration-dependent mobility matrix and logarithmic free energy. In particular we prove that there exists a unique solution for sufficiently smooth initial data. Further, we prove an error bound for a fully practical piecewise linear finite element approximation in one and two space dimensions. Finally numerical experiments with three components in one space dimension are presented

Finite element approximation of a model for phase separation of a multi-component alloy with nonsmooth free energy and a concentration dependent mobility matrix.

We consider a model for phase separation of a multi-component alloy with nonsmooth free energy and a concentration dependent mobility matrix. In particular we prove that there exists a unique solution for sufficiently smooth initial data. Further, we prove an error bound for a fully practical piecewise linear finite element approximation in one and two space dimensions. Finally numerical experiments with three components in one space dimension are presented

An Error Bound for the Finite Element Approximation of a Model for Phase Separation of a Multi-Component Alloy

An error bound is proved for a fully practical piecewise linear finite element approximation, using a backward-Euler time discretization, of a model for phase separation of a multi-component alloy. Numerical experiments with three components in one and two space dimensions are also presented

An improved error bound for a finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy.

Using the approach in [5] for analysing time discretization error and assuming more regularity on the initial data, we improve on the error bound derived in [2] for a fully practical piecewise linear finite element approximation with a backward Euler time discretization of a model for phase separation of a multi-component alloy with non-smooth free energy