Numerical analysis of the Cahn-Hilliard equation and approximation for the Hele-Shaw problem, Part II: Error analysis and convergence of the interface

Abstract

In this second part of the series, we focus on approximating the Hele-Shaw problem via the Cahn-Hilliard equation $u_t+\Delta (\varepsilon \Delta u -{\varepsilon}^{-1}f(u))=0$ as $\varepsilon \searrow 0$. The primary goal of this paper is to establish the convergence of the solution of the fully discrete mixed finite element scheme proposed in [21] 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]. Like in [20, 21], 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 [12], and establish a discrete counterpart of it for a linearized Cahn-Hilliard operator to handle the nonlinear term.