Error analysis of the unstructured mesh finite element method for the two-dimensional time-space fractional Schrödinger equation with a time-independent potential

Abstract

In this paper, error analysis of the unstructured mesh Galerkin finite element method for the two-dimensional time-space fractional Schrödinger equation with a time-independent potential defined on a finite domain is studied. The finite difference method is used to discretize the Caputo time fractional derivative, while the finite element method using unstructured mesh is used to deal with the Riesz fractional operators in space. Both the stability and convergence analysis of the numerical scheme are constructed. Numerical example is conducted to testify the validity of the proposed method. The conservation of the space fractional Schrödinger equation and the non-conservation of the time fractional Schrödinger equation in quantum mechanical system are achieved. This paper proposes an efficient numerical method as well as its theoretical analysis for the two-dimensional time-space fractional Schrödinger equation with time-independent potentials.

Keywords:

• Two-dimensional fractional Schrödinger equation
• Galerkin finite element method
• error analysis
• potential function
• finite difference method

2010 Mathematics Subject Classifications:

• 26A33
• 65M06
• 65M12
• 65M15
• 65M60

Acknowledgments

The authors would like to thank the editor and referees for their helpful comments which will improve the presentation 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 National Natural Science Foundation of China [grant numbers 11472161, 11771254, 11801221]; the Natural Science Foundation of Jiangsu Province [grant number BK20180586], and the Fundamental Research Funds for the Central Universities [grant number JUSRP11831].