Convergence analysis for a stabilized linear semi-implicit numerical scheme for the nonlocal Cahn–Hilliard equation

Li, Xiao; Qiao, Zhonghua; Wang, Cheng

doi:10.1090/mcom/3578

Convergence analysis for a stabilized linear semi-implicit numerical scheme for the nonlocal Cahn–Hilliard equation
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by Xiao Li, Zhonghua Qiao and Cheng Wang HTML | PDF

Math. Comp. 90 (2021), 171-188 Request permission

Abstract:

In this paper, we provide a detailed convergence analysis for a first order stabilized linear semi-implicit numerical scheme for the nonlocal Cahn–Hilliard equation, which follows from consistency and stability estimates for the numerical error function. Due to the complicated form of the nonlinear term, we adopt the discrete $H^{-1}$ norm for the error function to establish the convergence result. In addition, the energy stability obtained by Du et al., [J. Comput. Phys. 363 (2018), pp. 39–54] requires an assumption on the uniform $\ell ^\infty$ bound of the numerical solution, and such a bound is figured out in this paper by conducting the higher order consistency analysis. Taking the view that the numerical solution is indeed the exact solution with a perturbation, the error function is $\ell ^\infty$ bounded uniformly under a loose constraint of the time step size, which then leads to the uniform maximum-norm bound of the numerical solution.

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