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


Authors: Xiao Li, Zhonghua Qiao and Cheng Wang
Journal: Math. Comp. 90 (2021), 171-188
MSC (2010): Primary 35Q99, 65M12, 65M15, 65M70
DOI: https://doi.org/10.1090/mcom/3578
Published electronically: September 14, 2020
MathSciNet review: 4166457
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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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Additional Information

Xiao Li
Affiliation: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
ORCID: 0000-0003-3598-9077
Email: xiao1li@polyu.edu.hk

Zhonghua Qiao
Affiliation: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
MR Author ID: 711384
Email: zqiao@polyu.edu.hk

Cheng Wang
Affiliation: Department of Mathematics, The University of Massachusetts, North Dartmouth, Massachusetts
MR Author ID: 652762
Email: cwang1@umassd.edu

DOI: https://doi.org/10.1090/mcom/3578
Keywords: Nonlocal Cahn--Hilliard equation, Stabilized linear scheme, convergence analysis, higher order consistency expansion.
Received by editor(s): November 13, 2018
Received by editor(s) in revised form: January 5, 2020, and March 16, 2020
Published electronically: September 14, 2020
Additional Notes: The first author’s work was partially supported by NSFC grant 11801024.
The second author’s work was partially supported by the Hong Kong Research Council GRF grants 15325816 and 15300417.
The third author’s work was partially supported by NSF grant NSF DMS-1418689.
The second author is the corresponding author.
Article copyright: © Copyright 2020 American Mathematical Society