A new Crank-Nicolson finite element method for the time-fractional subdiffusion equation

In this paper, a new Crank-Nicolson finite element method for the time-fractional subdiffusion equation is developed, in which a novel time discretization called the modified L1 method is used to discretize the Riemann-Liouville fractional derivative. The present method is unconditionally stable and convergent of order O(τ1+β+hr+1), where β∈(0,1), τ and h are the step sizes in time and space, respectively, and r is the degree of the piecewise polynomial space. The derived method is reduced to the classical Crank-Nicolson method when β→1. The new time discretization is also used to solve the fractional cable equation. And the unconditional stability and convergence are given. Numerical examples are provided which support the theoretical analysis. The comparison with the existing methods are also given, which shows good performances of the present methods.