Charles M. Elliott

On the Cahn-Hilliard equation with degenerate mobility

An existence result for the Cahn–Hilliard equation with a concentration dependent diffusional mobility is presented. In particular, the mobility is allowed to vanish when the scaled concentration takes the values

Computation of geometric partial differential equations and mean curvature flow

\pm 1

The Cahn–Hilliard equation with a concentration dependent mobility: motion by minus the Laplacian of the mean curvature

, and it is shown that the solution is bounded by 1 in magnitude. Finally, applications of our method to other degenerate fourth-order parabolic equations are discussed.

A second order splitting method for the Cahn-Hilliard equation

This review concerns the computation of curvature-dependent interface motion governed by geometric partial differential equations. The canonical problem of mean curvature flow is that of finding a surface which evolves so that, at every point on the surface, the normal velocity is given by the mean curvature. In recent years the interest in geometric PDEs involving curvature has burgeoned. Examples of applications are, amongst others, the motion of grain boundaries in alloys, phase transitions and image processing. The methods of analysis, discretization and numerical analysis depend on how the surface is represented. The simplest approach is when the surface is a graph over a base domain. This is an example of a sharp interface approach which, in the general parametric approach, involves seeking a parametrization of the surface over a base surface, such as a sphere. On the other hand an interface can be represented implicitly as a level surface of a function, and this idea gives rise to the so-called level set method. Another implicit approach is the phase field method, which approximates the interface by a zero level set of a phase field satisfying a PDE depending on a new parameter. Each approach has its own advantages and disadvantages. In the article we describe the mathematical formulations of these approaches and their discretizations. Algorithms are set out for each approach, convergence results are given and are supported by computational results and numerous graphical figures. Besides mean curvature flow, the topics of anisotropy and the higher order geometric PDEs for Willmore flow and surface diffusion are covered.

Finite element methods for surface PDEs

We show by using formal asymptotics that the zero level set of the solution to the Cahn–Hilliard equation with a concentration dependent mobility approximates to lowest order in ɛ. an interface evolving according to the geometric motion, (where V is the normal velocity, Δ 8 is the surface Laplacian and κ is the mean curvature of the interface), both in the deep quench limit and when the temperature θ is where є 2 is the coefficient of gradient energy. Equation (0.1) may be viewed as motion by surface diffusion, and as a higher-order analogue of motion by mean curvature predicted by the bistable reaction-diffusion equation.

The Cahn-Hilliard Model for the Kinetics of Phase Separation

SummaryA semi-discrete finite element method requiring only continuous element is presented for the approximation of the solution of the evolutionary, fourth order in space, Cahn-Hilliard equation. Optimal order error bounds are derived in various norms for an implementation which uses mass lumping. The continuous problem has an energy based Lyapunov functional. It is proved that this property holds for the discrete problem.

Spinodal decomposition in Fe-Cr alloys: Experimental study at the atomic level and comparison with computer models—I. Introduction and methodology

In this article we consider finite element methods for approximating the solution of partial differential equations on surfaces. We focus on surface finite elements on triangulated surfaces, implicit surface methods using level set descriptions of the surface, unfitted finite element methods and diffuse interface methods. In order to formulate the methods we present the necessary geometric analysis and, in the context of evolving surfaces, the necessary transport formulae. A wide variety of equations and applications are covered. Some ideas of the numerical analysis are presented along with illustrative numerical examples.

The global dynamics of discrete semilinear parabolic equations

In this paper we consider the Cahn-Hilliard mathematical continuum model of spinodal decomposition (or phase separation) of a binary alloy. The phenomenological model is derived in section one. The existence theory for the Cahn-Hilliard equation is reviewed in section two. Various aspects and generalizations are surveyed in section three. A finite element approximation is studied in section four and, in particular, two fully discrete schemes are shown to possess Lyapunov functionals. Finally in section five some numerical simulations are described.

Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation

Abstract A three-part series of papers is presented concerning the atomic scale analysis of spinodal decomposition in Fe-Cr alloys. This first part deals with the experimental techniques and computer simulations, the second part discusses the dynamics of early stage phase separation, and the third part describes the morphological and structural characterization of spinodal microstructures. In this first paper, three-dimensional reconstructions of the atomic structure of a series of thermally aged Fe-Cr alloys are shown. Two methods for computer simulation of the decomposition process are described. The first is an atomistic simulation based on the Monte Carlo algorithm and the second is a numerical solution to the Cahn—Hilliard—Cook theory. The three-dimensional atomic scale structures resulting from decomposition within the low temperature miscibility gap are reconstructed. It is shown that both models generate microstructures which are qualitatively similar to those observed experimentally.

Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy

A class of scalar semilinear parabolic equations possessing absorbing sets, a Lyapunov functional, and a global attractor are considered. The gradient structure of the problem implies that, provided all steady states are isolated, solutions approach a steady state as