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 LI 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.

Automated summary

In this study, a fast finite element scheme is proposed for a hidden-memory variable-order time-fractional diffusion equation. By approximating the hidden-memory variable-order fractional derivative, the computational cost of generating coefficients is reduced. Error estimates for the scheme are proven using techniques from the analysis of L1 methods. Additionally, a fast divide and conquer algorithm is introduced to reduce the complexity of solving linear systems.