Two efficient Galerkin finite element methods for the modified anomalous subdiffusion equation

ABSTRACT

In this paper, we consider the numerical approximation of the modified anomalous subdiffusion model which involves the Riemann–Liouville derivatives in time. We propose two robust fully discrete finite element methods by employing the piecewise linear Galerkin finite element method in space and the convolution quadrature in time generated by the backward Euler and the second-order backward difference methods. The error estimates for semidiscrete and fully discrete schemes are investigated with respect to the data regularity. Furthermore, we numerically compare our numerical schemes with a Crank–Nicolson finite element method to illustrate the efficiency of our methods and confirm the theoretical results.

Keywords:

• Modified anomalous subdiffusion equation
• finite element method
• convolution quadrature
• error estimate
• data regularity

2010 Mathematics Subject Classifications:

• 65M60
• 65M12
• 65M15

Acknowledgments

The author wishes to thank the referees for their constructive comments and suggestions, which greatly improved the quality of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Guangxi Natural Science Foundation [grant numbers 2018GXNSFAA138121, 2018GXNSFBA281020] and the Doctoral Starting up Foundation of Guilin University of Technology [grant number GLUTQD2016044].