Superconvergence analysis of an H¹-Galerkin mixed finite element method for two-dimensional multi-term time fractional diffusion equations

ABSTRACT

In this paper, numerical approximation for two-dimensional (2D) multi-term time fractional diffusion equation is considered. By virtue of properties of bilinear element, Raviart–Thomas element and $L1$ approximation, an $H^1$-Galerkin mixed finite element fully discrete approximate scheme is established for 2D multi-term time fractional diffusion equation. And then, unconditionally stable of the approximate scheme is rigourously testified by dealing with fractional derivative skilfully. At the same time, superclose results for the original variable u in $H^1$-norm and the flux $\overrightarrow{q}=\mathrm{\nabla }u$ in $H(\mathrm{div},\mathrm{\Omega })$-norm are derived. Furthermore, the global superconvergence results for u in $H^1$-norm are deduced by the interpolation postprocessing operator. Finally, numerical results demonstrate that the approximate scheme provides a valid and efficient way for solving 2D multi-term time fractional diffusion equation.

KEYWORDS:

• Multi-term time fractional diffusion equation
• $H^1$-Galerkin mixed finite element
• $L1$ approximation
• stability
• superclose and superconvergence

2010 AMS SUBJECT CLASSIFICATIONS:

• 65N30
• 35R11

Disclosure statement

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

Additional information

Funding

The corresponding author (Yanmin Zhao) appreciates support of the National Natural Science Foundation of China (No. 11101381) and Outstanding Young Talents Training Plan by Xuchang University (No. [2015] 3).