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

Abstract

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.

22 pages, 12 figures