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

Na Li, Yongchao Zhao|arXiv (Cornell University)|Feb 25, 2026

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TL;DR

The paper develops a linear, energy-stable SAV-based finite element scheme for the Cahn-Hilliard equation, and proves unconditional energy stability and optimal-order convergence in time and space with numerical validation.

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.

Motivation & Objective

• Motivate and analyze a numerically efficient scheme for the Cahn-Hilliard equation that preserves energy dissipation at the discrete level.
• Introduce a SAV-based reformulation to linearize nonlinearities while maintaining stability.
• Establish unconditional energy stability and unique solvability for the fully discrete scheme.
• Derive a priori error estimates showing optimal first-order time and optimal second-order spatial convergence.
• Validate theoretical results with numerical experiments confirming convergence rates.

Proposed method

• Reformulate the CH equation using a scalar auxiliary variable r(t) to obtain a SAV-based system.
• Apply backward Euler time discretization and linear finite elements in space to achieve a linear system at each step.
• Prove unconditional energy stability via a discrete energy law and show unique solvability of the fully discrete scheme.
• Introduce an elliptic projection to derive rigorous error estimates for φ, μ, and r in the H1-norm.
• Establish convergence results showing 1st-order in time and 2nd-order in space for the phase variable and chemical potential.

(a) Variation of the mass with time

Experimental results

Research questions

• RQ1 Does the SAV-based fully discrete scheme for the Cahn-Hilliard equation remain unconditionally energy stable for all time steps?
• RQ2 What are the a priori error estimates for the phase field φ, chemical potential μ, and auxiliary variable r in the SAV finite element framework?
• RQ3 Under suitable regularity, what is the order of convergence in time and space for the SAV-FE scheme?
• RQ4 How does the elliptic projection technique contribute to the convergence analysis of the SAV-based scheme?

Key findings

• The SAV finite element scheme is unconditionally energy stable with a discrete energy that monotonically decreases in time.
• The scheme is uniquely solvable at each time step due to the linear structure of the discretization.
• Optimal-order convergence is established: first-order in time and optimal-order in space in the H1 norm for φ and μ, and suitable convergence for r.
• An elliptic projection is used to achieve rigorous error estimates and to separate projection and discretization errors.
• Numerical experiments corroborate the theoretical convergence rates and stability of the proposed method.

(b) Variation of the total free energy with time