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

Na Li, Yongchao Zhao

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.

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

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.
Paper Structure (8 sections, 3 theorems, 74 equations, 2 figures, 2 tables)

This paper contains 8 sections, 3 theorems, 74 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Mathematical Formulation and Preliminaries
  3. The Fully Discrete Scheme and Error Estimates
  4. Existence and Uniqueness of Solution and Unconditional Energy Stability
  5. Introduction of a Projection
  6. Error Estimates
  7. Algorithm Design
  8. Numerical Examples and Results

Key Result

Theorem 3.1

The system f1--f3 is unconditionally uniquely solvable and is unconditionally stable with respect to the discrete total free energy Specifically, the following energy inequality holds: which demonstrates the monotonic decay of the discrete energy.

Figures (2)

  • Figure 4.1: Temporal Evolution of Mass and Energy
  • Figure 4.2: Evolution of the phase field variable with time

Theorems & Definitions (3)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3