Numerical schemes for a three component Cahn-Hilliard model

Abstract

Abstract.In this article, we investigate numerical schemes for solving a three component Cahn-Hilliard model. The space discretization is performed by using a Galerkin formulation and the finite element method. Concerning the time discretization, the main difficulty is to write a scheme ensuring, at the discrete level, the decrease of the free energy and thus the stability of the method. We study three different schemes and prove existence and convergence theorems. Theoretical results are illustrated by various numerical examples showing that the new semi-implicit discretization that we propose seems to be a good compromise between robustness and accuracy.Mathematics Subject Classification. 35K55, 65M60, 65M12, 76T30.

An unconditionally stable uncoupled scheme for a triphasic Cahn–Hilliard/Navier–Stokes model

“…The existence of weak solutions for problem (1.1) together with initial and Neumann boundary conditions (for order parameters c i and chemical potentials µ i ) was proved in [4] (see [7] for an alternative proof based a numerical schemes) in 2D and 3D under the following general assumptions:…”

“…We then explain, in the next two paragraphs, the reasoning leading to the discretization of the coupling terms before writing the complete scheme in the last paragraph. For more details on the time discretizations of the triphasic Cahn-Hilliard model, the reader may refer to the article [7] (and references therein). Several articles in the literature are devoted to the study of discretizations of the Navier-Stokes equation: we refer in particular to the articles [15] and [21] which deal with variable density models.…”

“…This kind of discretizations was presented and studied in reference [7]. This is out the scope of the article.…”

“…The assumption (2.1) is a consistency assumption and the assumption (2.2) is the counterpart of the polynomial growth assumption (1.7) on F . Many possible choices for the discretization of the term d F i were presented in [7]. For instance, we consider in numerical tests of this article (section 4) the following expression:…”

“…The topological degree theory allows to deduce the existence of solutions for the non linear problem from a priori estimates which are in our case deduced from the energy equality (3.1) proved in proposition 3.1. Such a methodology was used for the approximation of the triphasic Cahn-Hilliard model in [7]. We only give here the main steps of the proof.…”