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

Authors：Guo, Jing; Wang, Cheng^*; Wise, Steven M.; Yue, Xingye

Source：Journal of Computational and Applied Mathematics, 2021, 388: 113300.

DOI：10.1016/j.cam.2020.113300

• Cahn-Hilliard equation
• Uniform-in-time H-m0 stability
• Linearized spectrum estimate
• Error analysis with an improved convergence constant

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(CT epsilon(-m0)), with m(0) a positive integer and the interface width parameter epsilon 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/epsilon only in a polynomial order, instead of the exponential order.

• Institution
  苏州大学