The Galerkin finite element method for a multi-term time-fractional diffusion equation

Jin, Bangti; Lazarov, Raytcho; Liu, Yikan; Zhou, Zhi

The Galerkin finite element method for a multi-term time-fractional diffusion equation

License

http://creativecommons.org/licenses/by/3.0/

Type

Article

Authors

Jin, Bangti
Lazarov, Raytcho
Liu, Yikan
Zhou, Zhi

KAUST Grant Number

KUS-C1-016-04

Date

2015-01

Abstract

© 2014 The Authors. We consider the initial/boundary value problem for a diffusion equation involving multiple time-fractional derivatives on a bounded convex polyhedral domain. We analyze a space semidiscrete scheme based on the standard Galerkin finite element method using continuous piecewise linear functions. Nearly optimal error estimates for both cases of initial data and inhomogeneous term are derived, which cover both smooth and nonsmooth data. Further we develop a fully discrete scheme based on a finite difference discretization of the time-fractional derivatives, and discuss its stability and error estimate. Extensive numerical experiments for one- and two-dimensional problems confirm the theoretical convergence rates.

Citation

Jin B, Lazarov R, Liu Y, Zhou Z (2015) The Galerkin finite element method for a multi-term time-fractional diffusion equation. Journal of Computational Physics 281: 825–843. Available: http://dx.doi.org/10.1016/j.jcp.2014.10.051.

Acknowledgements

The authors are grateful to the anonymous referees for their constructive comments. The research of B. Jin has been supported by NSF Grant DMS-1319052, R. Lazarov was supported in parts by NSF Grant DMS-1016525 and also by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST), and Y. Liu was supported by the Program for Leading Graduate Schools, MEXT, Japan.