Numerical Analysis of the Cahn-hilliard Equation and Approximation for the Hele-shaw Problem, Part Ii: Error Analysis and Convergence of the Interface

In this second part of the series, we focus on approximating the Hele-Shaw problem via the Cahn-Hilliard equation ut + ∆(ε∆u − εf(u)) = 0 as ε ↘ 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 HeleShaw (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]. 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.