Geodesic
Numerical analysis of the Cahn-Hilliard equation and approximation for the Hele-Shaw problem
Xiaobing Feng1 ; Andreas Prohl2
1University of Tennessee, Knoxville, USA
2Universität Tübingen, Germany
Interfaces and free boundaries, Tome 7 (2005) no. 1, pp. 1-28

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DOI

This paper concerns numerical approximations for the Cahn–Hilliard equation ut​+Δ(εΔu−ε−1f(u))=0 and its sharp interface limit as ε↘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 [29] 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∞)-error estimate, and making full use of the convergence result of [2]. The cruxes of the analysis are to establish stability estimates for the discrete solutions, use a spectrum estimate result of Alikakos and Fusco [3] and Chen [15], and establish a discrete counterpart of it for a linearized Cahn–Hilliard operator to handle the nonlinear term.
DOI : 10.4171/ifb/111
Classification : 65-XX, 76-XX, 92-XX, 00-XX
Mots-clés : Cahn-Hilliard equation, Hele-Shaw (Mullins-Sekerka) problem, phase transition, biharmonic problem, fully discrete mixed finite element method, Ciarlet-Raviart element

Xiaobing Feng  1   ; Andreas Prohl  2

1 University of Tennessee, Knoxville, USA
2 Universität Tübingen, Germany 
Xiaobing Feng; Andreas Prohl. Numerical analysis of the Cahn-Hilliard equation and approximation for the Hele-Shaw problem. Interfaces and free boundaries, Tome 7 (2005) no. 1, pp. 1-28. doi: 10.4171/ifb/111
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