  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.
