# Numerical analysis of the Cahn-Hilliard equation and approximation for the Hele-Shaw problem

### Andreas Prohl

Universität Tübingen, Germany### Xiaobing H. Feng

University of Tennessee, Knoxville, USA

## Abstract

This paper concerns numerical approximations for the Cahn-Hilliard equation $u_t+\Delta (\varepsilon \Delta u -{\varepsilon}^{-1}f(u))=0$ and its sharp interface limit as $\varepsilon \searrow 0$, known as the Hele-Shaw problem. The primary goal of this paper is to establish the convergence of the solution of the fully discrete mixed finite element scheme proposed in \cite{XA2} to the solution of the Hele-Shaw (Mullins-Sekerka) problem, provided that the Hele-Shaw (Mullins-Sekerka) problem has a global (in time) classical solution. This is accomplished by establishing some improved a priori solution and error estimates, in particular, an $L^\infty(L^\infty)$-error estimate, and making full use of the convergence result of \cite{alikakos94}. The cruxes of the analysis are to establish stability estimates for the discrete solutions, use a spectrum estimate result of Alikakos and Fusco \cite{alikakos93} and Chen \cite{chen94}, and establish a discrete counterpart of it for a linearized Cahn-Hilliard operator to handle the nonlinear term.

## Cite this article

Andreas Prohl, Xiaobing H. Feng, Numerical analysis of the Cahn-Hilliard equation and approximation for the Hele-Shaw problem. Interfaces Free Bound. 7 (2005), no. 1, pp. 1–28

DOI 10.4171/IFB/111