Error Analysis Of Energy Stable Weak Galerkin Schemes For The Allen-Cahn Equation

Document Type

Article

Publication Date

3-1-2026

School

Mathematics and Natural Sciences

Abstract

In this study, we explore energy-stable weak Galerkin finite element methods for solving the Allen–Cahn equation, a widely used mathematical model for phase separation. To this end, we introduce a new energy functional consistent with the weak Galerkin framework. We then propose two novel schemes: a first-order conditionally energy-stable scheme and a second-order unconditionally energy-stable scheme. We rigorously analyze the energy stability and convergence properties of these methods. Through numerical experiments, we verify the spatial and temporal error estimates as well as the corresponding convergence rates. Furthermore, we apply these methods to mean curvature flow and phase-field models to demonstrate their energy stability and accuracy.

Publication Title

Advances in Computational Science and Engineering

Volume

7

First Page

48

Last Page

72

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