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

TL;DR

This work develops invariant energy quadratization (IEQ) finite element methods for the Cahn-Hilliard and Allen-Cahn gradient flows. It systematically analyzes three first-order time-discretization variants that differ in where the IEQ intermediate function is approximated, including a fully coupled FE solve, a continuous-space approach, and a projected continuous-space approach, all yielding linear, unconditionally energy-stable schemes. Existence/uniqueness results, mass conservation for CH, and energy-dissipation properties are established, complemented by extensive numerical tests that confirm accuracy, efficiency, and stability across CH and AC, including 2D and 3D validations. The methods enable reliable long-time simulations of phase-field models by preserving the underlying energy laws while offering computationally attractive implementations; potential extensions include error analysis, adaptivity, and higher-order time discretizations.

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.

Figures (15)

• Figure 1: $\mathbf{Example\ \ref{['2e1a']}}$, $L_2$ errors and convergent rates for the CH equation, Left: Time accuracy test ($P_{2}$ element, $T=1$, uniform mesh with $(h_x, h_y)=(\frac{4 \pi}{200},\frac{4 \pi}{200})$); Right: Spatial accuracy test ($\Delta t=1e^{-6},\, T=1e^{-3}$).
• Figure 2: $\mathbf{Example\ \ref{['exmS3']}\ (CH, T=8e-5)}$, First order scheme; Left: BDF1-IEQ-FEM1 scheme (\ref{['BDF1-IEQ-FEM1']}), Middle: BDF1-IEQ-FEM2 scheme (\ref{['BDF1-IEQ-FEM2']}), Right: BDF1-IEQ-FEM3 scheme (\ref{['BDF1-IEQ-FEM3']}); First row: Numerical solution; Second row: Time evolution of the discrete energy.
• Figure 3: $\mathbf{Example\ \ref{['cexm2-n']}\ (CH)}$, Effect of constant B on the discrete energy, Left: BDF1-IEQ-FEM2 scheme (\ref{['BDF1-IEQ-FEM2']}), Right: BDF1-IEQ-FEM3 scheme (\ref{['BDF1-IEQ-FEM3']}).
• Figure 4: $\mathbf{Example\ \ref{['cexm2-n']}\ (CH)}$, numerical solutions, First and second line: BDF2-IEQ-FEM2 scheme (\ref{['BDF2-IEQ-FEM2']}), Third and forth line: BDF2-IEQ-FEM3 scheme (\ref{['BDF2-IEQ-FEM3']}).
• Figure 5: $\mathbf{Example\ \ref{['cexm2-n']}\ (CH)}$, Left: The discrete energy, Right: the total mass.

Theorems & Definitions (38)

• Lemma 2.1
• proof
• Lemma 2.2
• Theorem 2.1
• proof
• Remark 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Theorem 2.2