Fast algorithm based on TT-M FE system for space fractional Allen-Cahn equations with smooth and non-smooth solutions

JOURNAL OF COMPUTATIONAL PHYSICS

卷:379

Article

Yin, Baoli¹  Liu, Yang¹  Li, Hong¹  He, Siriguleng¹ 

[1] Inner Mongolia Univ, Sch Math Sci, Hohhot 010021, Peoples R China

关键词: Fast algorithm based on TT-M FE system;    Space fractional Allen-Cahn equations;    Stability;    A priori error estimates;    CPU time;    Non-smooth data;   

DOI  :  10.1016/j.jcp.2018.12.004

来源: Elsevier

【 摘 要 】

In this article, a fast algorithm based on time two-mesh (TT-M) finite element (FE) scheme, which aims at solving nonlinear problems quickly, is considered to numerically solve the nonlinear space fractional Allen-Cahn equations with smooth and non-smooth solutions. The implicit second-order theta scheme containing both implicit Crank-Nicolson scheme and second-order backward difference method is applied to time direction, a fast TT-M method is used to increase the speed of calculation, and the FE method is developed to approximate the spacial direction. The TT-M FE algorithm includes the following main computing steps: firstly, a nonlinear implicit second-order theta FE scheme on the time coarse mesh tau(c) is solved by a nonlinear iterative method; secondly, based on the chosen initial iterative value, a linearized FE system on time fine mesh tau < tau(c) is solved, where some useful coarse numerical solutions are found by the Lagrange's interpolation formula. The analysis for both stability and a priori error estimates are made in detail. Finally, three numerical examples with smooth and non-smooth solutions are provided to illustrate the computational efficiency in solving nonlinear partial differential equations, from which it is easy to find that the computing time can be saved. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free