An adaptive discontinuous finite volume element method for the Allen-Cahn equation

• Published in: Advances in computational mathematics Vol. 49; no. 4
• Main Authors: Li, Jian, Zeng, Jiyao, Li, Rui
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.08.2023; Springer Nature B.V
• Subjects: Computational mathematics; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Dynamic stability; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Numerical methods; Phase transitions; Visualization; Error estimates; Discontinuous finite volume element method; The Allen-Cahn equation; Discrete energy dissipation
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-023-10031-5

In this paper, the discontinuous finite volume element method (DFVEM) is considered to solve the Allen-Cahn equation which contains strong nonlinearity. The method is based on the DFVEM in space and the backward Euler method in time. The energy stability and unique solvability of the proposed fully discrete scheme are derived. The error estimates for the semi-discrete and fully discrete scheme are also established. A series of numerical experiments verify the efficiency of the proposed numerical method. The results show that our method can not only accurately capture the dynamic information of the phase transition, but also ensure the stability of the system during long-term numerical simulations.