Numerical analysis of the Cahn-Hilliard equation and approximation for the Hele-Shaw problem, Part I: Error analysis under minimum regularities

Abstract

In this first part of a series, we propose and analyze, under minimum regularity assumptions, a semi-discrete (in time) scheme and a fully discrete mixed finite element scheme for the Cahn-Hilliard equation $u_t+\Delta (\varepsilon \Delta u -{\varepsilon}^{-1}f(u))=0$ arising from phase transition in materials science, where $\vepsi$ is a small parameter known as an ``interaction length". The primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical methods, in particular, by focusing on the dependence of the error bounds on $\varepsilon$. Quasi-optimal order error bounds are shown for the semi-discrete and fully discrete schemes under different constraints on the mesh size $h$ and the local time step size $k_m$ of the stretched time grid, and minimum regularity assumptions on the initial function $u_0$ and domain $\Omega$. In particular, all our error bounds depend on $\frac{1}{\varepsilon}$ only in some lower polynomial order for small $\varepsilon$. The cruxes of the analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of Alikakos and Fusco [3] and Chen [15], and to establish a discrete counterpart of it for a linearized Cahn-Hilliard operator to handle the nonlinear term on the stretched time grid. It is this polynomial dependency of the error bounds that paves the way for us to establish convergence of the numerical solution to the solution of the Hele-Shaw (Mullins-Sekerka) problem (as $\varepsilon \searrow 0$) in Part II \cite{XA3} of the series.