Non-smooth Free Energy Research Articles

On fully practical finite element approximations of degenerate Cahn-Hilliard systems

We consider a model for phase separation of a multi-component alloy with non-smooth free energy and a degenerate mobility matrix. In addition to showing well-posedness and stability bounds for our approximation, we prove convergence in one space dimension. Furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed. We discuss also how our approximation has to be modified in order to be applicable to a logarithmic free energy. Finally numerical experiments with three components 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 with a concentration dependent mobility matrix

Using a slightly different discretization scheme in time and adapting the approach in Nochetto et al. (1998) for analysing the time discretization error in the backward Euler method, we improve on the error bounds derived in (i) Barrett and Blowley (1998) and (ii) Barrett and Blowey (1999c) for a fully practical piecewise linear finite element approximation of a model for phase separation of a multi-component alloy with a concentration dependent mobility matrix and (i) a logarithmic free energy, and (ii) a non-smooth free energy (the deep quench limit); respectively. Moreover, the improved error bound in the deep quench limit is optimal. Numerical experiments with three components illustrating the above error bounds are also presented.

An optimal 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 reg- ularity on the initial data, we improve on the error bound derived in (2) for a fully practical piecewise linear nite element approximation with a backward Euler time discretization of a model for phase separation of a multi-component alloy with non-smooth free energy. AMS Subject Classication. 65M60, 65M15, 35K55, 35K35.

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.

Finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy

A model for the phase separation of a multi-component alloy with non-smooth free energy is considered. An error bound is proved for a fully practical piecewise linear finite element approximation using a backward Euler time discretization. An iterative scheme for solving the resulting nonlinear algebraic system is analysed. Finally numerical experiments with three components in one and two space dimensions are presented. In the one dimensional case we compare some steady states obtained numerically with the corresponding stationary solutions of the continuous problem, which we construct explicitly.

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.

The Cahn–Hilliard gradient theory for phase separation with non-smooth free energy Part I: Mathematical analysis

A mathematical analysis is carried out for the Cahn–Hilliard equation where the free energy takes the form of a double well potential function with infinite walls. Existence and uniqueness are proved for a weak formulation of the problem which possesses a Lyapunov functional. Regularity results are presented for the weak formulation, and consideration is given to the asymptotic behaviour as the time becomes infinite. An investigation of the associated stationary problem is undertaken proving the existence of a nontrivial stationary solution and further regularity results for any stationary solution. Stationary solutions are constructed in one and two dimensions; a formula for the number of stationary solutions in one dimension is derived. It is then natural to study the asymptotic behaviour as the phenomenological parameter λ→0, the main result being that the interface between the two phases has minimal area.