A dimension reduction method of two‐grid finite element solution coefficient vectors for the Allen–Cahn equation

• Published in: Mathematical methods in the applied sciences Vol. 48; no. 1; pp. 678 - 698
• Main Authors: Li, Yuejie, Teng, Fei, Zeng, Yihui, Luo, Zhendong
• Format: Journal Article
• Language: English
• Published: Freiburg Wiley Subscription Services, Inc 15.01.2025
• Subjects: Errors; Proper Orthogonal Decomposition; reduced‐dimension extrapolated two‐grid Crank–Nicolson finite element method; Stability; the nonlinear Allen–Cahn equation; two‐grid Crank–Nicolson finite element method; unconditional stability and errors
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.10350

This paper mainly focuses on the dimension reduction of unknown finite element (FE) solution coefficient vectors in two‐grid Crank–Nicolson FE (TGCNFE) method for the nonlinear Allen–Cahn equation. For this purpose, a new TGCNFE method for the nonlinear Allen–Cahn equation is first developed and the unconditional stability and errors of TGCNFE solutions are analyzed. Thereafter, the most important thing is to lower the dimension of the unknown FE solution coefficient vectors by a proper orthogonal decomposition, to establish a new reduced‐dimension extrapolated TGCNFE (RDETGCNFE) method, and to analyze the unconditional stability and errors of RDETGCNFE solutions. Moreover, the correctness of our theoretical results is validated by some numerical experiments.