Electronic Journal of Differential Equations,
Vol. 2019 (2019), No. 76, pp. 1-17.

Title: Crank-Nicolson Legendre spectral approximation for space-fractional
       Allen-Cahn equation

Authors: Wenping Chen (Beijing Univ. of Aeronautics and Astronautics, Beijing, China)
         Shujuan Lu   (Beijing Univ. of Aeronautics and Astronautics, Beijing, China)
         Hu Chen      (Beijing Computational Science Research Center, Beijing, China)
         Haiyu Liu    (Beijing Univ. of Aeronautics and Astronautics, Beijing, China)

Abstract:
 In this article, we consider spectral methods for solving the
 initial-boundary value problem of the space fractional-order 
 Allen-Cahn equation.
 A fully discrete scheme based on the modified Crank-Nicolson scheme in
 time and the Legendre spectral method in space is established.
 The existence and uniqueness of the fully discrete scheme are derived,
 and the stability and convergence analysis of the fully discrete scheme
 are proved rigorously.
 By constructing a fractional duality argument, the corresponding optimal
 error estimates in $L^2$ and $H^\alpha$ norm are derived, respectively.
 Also, numerical experiments are performed to support the theoretical results.

Submitted August 28, 2018. Published May 31, 2019.
Math Subject Classifications: 35R11, 65M06, 65M70, 65M12.
Key Words: Space-fractional Allen-Cahn equation; Legendre spectral method; 
           modified Crank-Nicolson scheme; stability; convergence.