Error Analysis with Polynomial Dependence on ε-1 in SAV Methods for the Cahn-Hilliard Equation

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Original languageEnglish
Article number83
Journal / PublicationJournal of Scientific Computing
Volume101
Issue number3
Online published14 Nov 2024
Publication statusPublished - Dec 2024

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Abstract

The optimal error estimate that depends only on the polynomial degree of ε-1 is established for the temporal semi-discrete scheme of the Cahn-Hilliard equation based on the scalar auxiliary variable (SAV) formulation. The key to our analysis is converting the structure of the SAV time-stepping scheme back to a form compatible with the original format of the Cahn-Hilliard equation, which makes it feasible to use spectral estimates to handle the nonlinear term. Based on the transformation of the SAV numerical scheme, the optimal error estimate for the temporal semi-discrete scheme which depends only on the low polynomial order of ε-1 instead of the exponential order, is derived by using mathematical induction, spectral arguments, and the superconvergence properties of some nonlinear terms. Numerical examples are provided to illustrate the discrete energy decay property and validate our theoretical convergence analysis. © The Author(s) 2024.

Research Area(s)

  • Cahn-Hilliard equation, Energy decay, Error estimates, Polynomial order, SAV formulation, Spectral estimates

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