A two-level finite element method for the Allen–Cahn equation

ABSTRACT

We consider the fully implicit treatment for the nonlinear term of the Allen–Cahn equation. To solve the nonlinear problem efficiently, the two-level scheme is employed. We obtain the discrete energy law of the fully implicit scheme and two-level scheme with finite element method. Also, the convergence of the two-level method is presented. Finally, some numerical experiments are provided to confirm the theoretical analysis.

KEYWORDS:

• Finite element method
• two-level method
• Allen–Cahn equation
• stability and convergence
• numerical tests

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 65M12
• 65M55
• 65M60

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This study was subsidized by the National Natural Science foundation of China [grant number 11401466, 51236006, 11371288], China Postdoctoral Science Foundation [grant number 2013M540750], the Fundamental Research Funds for the Central Universities [grant number xjj2017127] and Ph.D. Programs Foundation of Ministry of Education of China [grant number 20130201120052].