A numerical approach for the Riesz space-fractional Fisher' equation in two-dimensions

ABSTRACT

In this paper, we consider a fully discrete finite element method (FEM) to solve the two-dimensional nonlinear Fisher' equation with Riesz fractional derivatives in space. This method is chiefly performed by using Crank–Nicolson discretization in conjunction with a linearized approach in time and FEM in space. The existence, uniqueness of the weak solution, and the numerical stability of the scheme are proved in great detail. The optimal error estimate computed by $L^2$-norm showed both in time and space is derived by introducing a fractional orthogonal projection. Moreover, several numerical examples are conducted on unstructured triangular meshes by a properly designed algorithm.

KEYWORDS:

• Crank–Nicolson
• finite element method
• Riesz space-fractional Fisher's equation
• convergence analysis

2010 AMS SUBJECT CLASSIFICATIONS:

• 35R11
• 65M60
• 65M12

Acknowledgements

The authors are very grateful for the reviewers' suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was supported by the National Natural Science Foundations of China [No. 11471262].