Convergence Analysis of a Linear, Unconditionally Energy-Stable SAV Finite Element Method for the Cahn-Hilliard Equation

TL;DR

This paper introduces a finite element method based on the SAV approach for the Cahn-Hilliard equation, proving its convergence, energy stability, and demonstrating its effectiveness through numerical tests.

Contribution

It provides a new unconditionally energy-stable SAV finite element scheme with proven optimal convergence for the Cahn-Hilliard equation.

Findings

Optimal-order convergence in time and space.

Unconditional energy stability of the scheme.

Numerical results confirm theoretical convergence rates.

Abstract

This paper proposes a finite element scheme, based on the Scalar Auxiliary Variable (SAV) approach, for the Cahn-Hilliard equation--a model that possesses significant physical relevance and a rich mathematical structure. A convergence analysis of the fully discrete scheme is conducted under suitable regularity assumptions, confirming optimal-order convergence in both time and space for the phase variable, chemical potential, and auxiliary variable in the H1-norm. Furthermore, the scheme is proven to be unconditionally energy stable. Finally, a numerical example is presented to demonstrate the effectiveness of the method and to confirm the theoretical convergence rates.