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

Abstract

This paper concerns numerical approximations for the Cahn–Hilliard equation $u_t+\Delta (\epsilon \Delta u-{\epsilon }^{-1}f(u))=0$ and its sharp interface limit as $\epsilon \searrow 0$, 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 [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 $L^{\infty }(L^{\infty })$-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.