JournalsifbVol. 7, No. 1pp. 1–28

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

This paper concerns numerical approximations for the Cahn-Hilliard equation ut+Δ(εΔuε1f(u))=0u_t+\Delta (\varepsilon \Delta u -{\varepsilon}^{-1}f(u))=0 and its sharp interface limit as ε0\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(L)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