A Stabilized bi-grid method for Allen Cahn equation in Finite Elements
This work addresses the computational cost of solving the Allen-Cahn equation, offering a more efficient numerical method for researchers in computational PDEs.
The paper proposes a bi-grid finite element method for the Allen-Cahn equation that separates solution into coarse and fine components, applying implicit time-stepping on the coarse grid and a simpler semi-implicit scheme on the fine grid. Numerical examples demonstrate improved stability and robustness, with significant reduction in computation time compared to fully implicit methods.
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.