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

This paper concerns numerical approximations for the Cahn-Hilliard equation u t + (u -1 f (u)) = 0 and its sharp interface limit as 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 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 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.