Error analysis of a mixed finite element method for the Cahn-Hilliard equation

Abstract

Summary.We propose and analyze a semi-discrete and a fully discrete mixed finite element method for the Cahn-Hilliard equation u t + (ε u − ε −1 f (u)) = 0, where ε > 0 is a small parameter. Error estimates which are quasi-optimal order in time and optimal order in space are shown for the proposed methods under minimum regularity assumptions on the initial data and the domain. In particular, it is shown that all error bounds depend on 1 ε only in some lower polynomial order for small ε. The cruxes of our analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of Alikakos and Fusco [2], and Chen [15] to prove a discrete counterpart of it for a linearized Cahn-Hilliard operator to handle the nonlinear term on a stretched time grid. The ideas and techniques developed in this paper also enable us to prove convergence of the fully discrete finite element solution to the solution of the Hele-Shaw (Mullins-Sekerka) problem as ε → 0 in [29].

A posterioriestimates for the Cahn–Hilliard equation with obstacle free energy

“…[16][17][18] and the references therein. A posteriori estimates for the Cahn-Hilliard equation with the smooth potential (1.4) have very recently been obtained in [19], where the estimates for a continuous in time semi-discrete approximation only depend on polynomial powers of γ −1 , a result which crucially depends on the spectral estimate from [13].…”

Energy-diminishing integration of gradient systems

“…Such problems arise from the spatial discretization of Allen-Cahn and Cahn-Hilliard partial differential equations (Barrett & Blowey, 2002;Feng & Prohl, 2004). We have the following direct consequence to Theorem 2.1.…”

The Willmore functional and instabilities in the Cahn-Hilliard equation

“…In Section 4 we perform numerical computations for (1.1) near transition solutions in one and two dimensions. We remark that in the past 20 years numerical approximations of the solutions of the Cahn-Hilliard equation -for purposes different from ours -have been studied by many authors, see [16] and [17] for further references. We use a semi-implicit approximation in time and finite elements for the space discretization in this paper.…”