An improved error analysis for a second-order numerical scheme for the Cahn-Hilliard equation

Abstract In this paper we present an error analysis for a second order accurate numerical scheme for the 2-D and 3-D Cahn-Hilliard (CH) equation, with an improved convergence constant. The unique solvability, unconditional energy stability, and a uniform-in-time H 2 stability of this numerical scheme have already been established. However, a standard error estimate gives a convergence constant in an order of exp ( C T e − m 0 ) , with m 0 a positive integer and the interface width parameter e being small, which comes from the application of discrete Gronwall inequality. To overcome this well-known difficulty, we apply a spectrum estimate for the linearized Cahn-Hilliard operator (Alikakos and Fusco, 1993; Chen, 1994; Feng and Prohl, 2004), perform a detailed numerical analysis, and get an improved estimate, in which the convergence constant depends on 1 e only in a polynomial order, instead of the exponential order.