We consider a generalisation of the Cahn–Hilliard equation that incorporates an elastic energy density which, being quasiconvex, incorporates microstructure formation on smaller length scales. We prove global existence of weak solutions in certain microstructural regimes in (one and) two dimensions and present sufficient conditions for uniqueness. Preliminary numerical computations to illustrat...

The Cahn-Hilliard equation has its origin in material sciences and serves as a model for phase separation and phase coarsening in binary alloys. A new approach in the class of fourth order inpainting algorithms is inpainting of binary images using the Cahn-Hilliard equation. We will present a generalization of this fourth order approach for grayvalue images. This is realized by using subgradien...

In this article, we are interested in the study of the asymptotic behavior, in terms of finite-dimensional attractors, of a generalization of the Cahn-Hilliard equation with a fidelity term (integrated over Ω\D instead of the entire domain Ω, D ⊂⊂ Ω). Such a model has, in particular, applications in image inpainting. The difficulty here is that we no longer have the conservation of mass, i.e. o...

In this paper, we consider the application of the local discontinuous Galerkin method for the Allen-Cahn/Cahn-Hilliard system. The method in this paper extends the local discontinuous Galerkin method in [10] to the more general application system which is coupled with the Allen-Cahn and Cahn-Hilliard equations. Similar energy stability result as that in [10] is presented. Numerical results for ...

Our aim in this paper is to study generalizations of the Allen-Cahn equation based on a modification of the Ginzburg-Landau free energy proposed in [25]. In particular, the free energy contains an additional term called Willmore regularization. We prove the existence, uniqueness and regularity of solutions, as well as the existence of the global attractor. Furthermore, we study the convergence ...

This paper develops a posteriori error estimates of residual type for conforming and mixed finite element approximations of the fourth order Cahn-Hilliard equation ut + ∆ ` ε∆u− ε−1f(u) ́ = 0. It is shown that the a posteriori error bounds depends on ε−1 only in some low polynomial order, instead of exponential order. Using these a posteriori error estimates, we construct an adaptive algorithm f...

Formation of modulated phase patterns can be modelized by a modi…ed Cahn-Hilliard equation which includes a non local term preventing the formation of macroscopic domains. Using stationnary solutions of the original Cahn-Hilliard equation as analytical ansatzs, we compute the thermodynamically stable period of a 1D modulated phase pattern. We …nd that the period scales like the power -1/3 of th...

We study finite element approximations of stochastic partial differential equations of Ginzburg-Landau type and the main paradigm considered in this paper is the stochastic Allen-Cahn model. We first demonstrate that the constructed stochastic finite element approximations are within an arbitrary level of tolerance from the corresponding one-dimensional stochastic partial differential equation;...

Keywords: Allen–Cahn equation Finite element method Operator splitting method Unconditionally stable scheme a b s t r a c t We present an unconditionally stable hybrid finite element method for solving the Allen– Cahn equation, which describes the temporal evolution of a non-conserved phase-field during the antiphase domain coarsening in a binary mixture. Its various modified forms have been ap...