Two-Grid Finite Element Method for the Time-Fractional Allen–Cahn Equation With the Logarithmic Potential

IF 2.1 3区数学 Q1 MATHEMATICS, APPLIED

Mathematical Methods in the Applied Sciences Pub Date : 2025-01-14 DOI:10.1002/mma.10704

Jiyu Zhang, Xiaocui Li, Wenyan Ma

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引用次数: 0

Abstract

In this paper, we propose a two-grid finite element method for solving the time-fractional Allen–Cahn equation with the logarithmic potential. Firstly, with the L1 method to approximate Caputo fractional derivative, we solve the fully discrete time-fractional Allen–Cahn equation on a coarse grid with mesh size $H$ and time step size $τ$ . Then, we solve the linearized system with the nonlinear term replaced by the value of the first step on a fine grid with mesh size $h$ and the same time step size $τ$ . We obtain the energy stability of the two-grid finite element method and the optimal order of convergence of the two-grid finite element method in the L2 norm when the mesh size satisfies $h = O (H^{2})$ . The theoretical results are confirmed by arithmetic examples, which indicate that the two-grid finite element method can keep the same convergence rate and save the CPU time.

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来源期刊

Mathematical Methods in the Applied Sciences 数学-应用数学

CiteScore

4.90

自引率

6.90%

发文量

798

审稿时长

6 months

期刊介绍： Mathematical Methods in the Applied Sciences publishes papers dealing with new mathematical methods for the consideration of linear and non-linear, direct and inverse problems for physical relevant processes over time- and space- varying media under certain initial, boundary, transition conditions etc. Papers dealing with biomathematical content, population dynamics and network problems are most welcome. Mathematical Methods in the Applied Sciences is an interdisciplinary journal: therefore, all manuscripts must be written to be accessible to a broad scientific but mathematically advanced audience. All papers must contain carefully written introduction and conclusion sections, which should include a clear exposition of the underlying scientific problem, a summary of the mathematical results and the tools used in deriving the results. Furthermore, the scientific importance of the manuscript and its conclusions should be made clear. Papers dealing with numerical processes or which contain only the application of well established methods will not be accepted. Because of the broad scope of the journal, authors should minimize the use of technical jargon from their subfield in order to increase the accessibility of their paper and appeal to a wider readership. If technical terms are necessary, authors should define them clearly so that the main ideas are understandable also to readers not working in the same subfield.