A Stabilized bi-grid method for Allen Cahn equation in Finite Elements

TL;DR

This paper introduces a bi-grid finite element scheme for the Allen-Cahn equation that stabilizes high-frequency modes, reducing computational effort while maintaining accuracy and robustness.

Contribution

A novel bi-grid scheme for the Allen-Cahn equation using dual FEM spaces and scale separation, enhancing stability and efficiency.

Findings

Significant reduction in computation time compared to fully implicit methods

Enhanced stability and robustness demonstrated through numerical examples

Effective stabilization of high-frequency components in the solution

Abstract

In this work, we propose a bi-grid scheme framework for the Allen-Cahn equation in Finite Element Method. The new methods are based on the use of two FEM spaces, a coarse one and a fine one, and on a decomposition of the solution into mean and fluctuant parts. This separation of the scales, in both space and frequency, allows to build a stabilization on the high modes components: the main computational effort is concentrated on the coarse space on which an implicit scheme is used while the fluctuant components of the fine space are updated with a simple semi-implicit scheme, they are smoothed without damaging the consistency. The numerical examples we give show the good stability and the robustness of the new methods. An important reduction of the computation time is also obtained when comparing our methods with fully implicit ones.