High-Order Numerical Methods for Solving Time Fractional Partial Differential Equations

In this paper we introduce a new numerical method for solving time fractional partial differential equation. The time discretization is based on Diethelm's method where the Hadamard finite-part integral is approximated by using the piecewise quadratic interpolation polynomials. The space discretization is based on the standard finite element method. The error estimates with the convergence order are proved in detail by using the argument developed recently by Lv and Xu (SIAM J Sci Comput 38:A2699-A2724, 2016), where and h denote the time and space step sizes, respectively. Numerical examples in both one- and two-dimensional cases are given.