Error estimates for the scalar auxiliary variable (SAV) scheme to the Cahn-Hilliard equation

TL;DR

This paper establishes optimal error estimates for a scalar auxiliary variable scheme applied to the Cahn-Hilliard equation, demonstrating improved dependence on parameters and validating results through numerical examples.

Contribution

The paper introduces a novel analysis converting SAV scheme structure to the original form, enabling optimal error estimates with dependence only on polynomial order.

Findings

Optimal error estimate depending on polynomial degree of ε^{-1}

Numerical examples confirm energy decay and convergence

Transformation of SAV scheme facilitates spectral analysis

Abstract

The optimal error estimate that depending only on the polynomial degree of $ \varepsilon^{-1}$ is established for the temporal semi-discrete scheme of the Cahn-Hilliard equation, which is based on the scalar auxiliary variable (SAV) formulation. The key to our analysis is to convert 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 $\varepsilon^{-1}$ 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.