High-accuracy finite element method for 2D time fractional diffusion-wave equation on anisotropic meshes

ABSTRACT

Employing finite element method in spatial direction and Crank–Nicolson scheme in temporal direction, a fully discrete scheme with high accuracy is established for a class of two-dimensional time fractional diffusion-wave equation with Caputo fractional derivative. Unconditional stability analysis of the approximate scheme is proposed. The spatial global superconvergence and temporal convergence of order $O(h^2+{\tau }^{3-\alpha })$ for the original variable in $H^1$-norm is presented by means of properties of bilinear element and interpolation postprocessing technique without Ritz projection, where h and τ are the step sizes in space and time, respectively. Finally, several numerical results are implemented to evaluate the efficiency of the theoretical results on both regular and anisotropic meshes.

KEYWORDS:

• Time fractional diffusion-wave equation
• finite element method
• Crank–Nicolson scheme
• stability
• convergence and superconvergence

2010 AMS SUBJECT CLASSIFICATION:

• 65N30
• 35R11

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [Nos. 11101381, 11771438] and Outstanding Young Talents Training Plan by Xuchang University.