Unconditionally energy stable IEQ-FEMs for the Cahn-Hilliard equation and Allen-Cahn equation

TL;DR

This paper develops several unconditionally energy-stable finite element methods for the Cahn-Hilliard and Allen-Cahn equations, ensuring stability and accuracy with rigorous proofs and numerical validation.

Contribution

It introduces new IEQ-FEM schemes with different function space placements for the intermediate function, providing unconditionally energy stability and mass conservation proofs.

Findings

Schemes are unconditionally energy-stable and mass-conserving.

Numerical experiments confirm accuracy and efficiency.

The methods outperform existing approaches in stability and solution quality.

Abstract

In this paper, we present several unconditionally energy-stable invariant energy quadratization (IEQ) finite element methods (FEMs) with linear, first- and second-order accuracy for solving both the Cahn-Hilliard equation and the Allen-Cahn equation. For time discretization, we compare three distinct IEQ-FEM schemes that position the intermediate function introduced by the IEQ approach in different function spaces: finite element space, continuous function space, or a combination of these spaces. Rigorous proofs establishing the existence and uniqueness of the numerical solution, along with analyses of energy dissipation for both equations and mass conservation for the Cahn-Hilliard equation, are provided. The proposed schemes' accuracy, efficiency, and solution properties are demonstrated through numerical experiments.