Numerical Analysis of a Fast Finite Element Method for a Hidden-Memory Variable-Order Time-Fractional Diffusion Equation

We investigate a fast finite element scheme to a hidden-memory variable-order time-fractional diffusion equation. Different from the traditional L1 methods, a fast approximation to the hidden-memory variable-order fractional derivative is derived to reduce the computational cost of generating coefficients from O ( N 2 ) to O ( N log N ) , where N refers to the number of time steps. We then develop different techniques from the analysis of L1 methods to prove error estimates for the corresponding fast fully-discrete finite element scheme. Furthermore, a fast divide and conquer algorithm is proposed to reduce the complexity of solving the linear systems from O ( M N 2 ) to O ( M N log 2 N ) where M stands for the spatial degree of freedom. Numerical experiments are presented to substantiate the theoretical results.