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.

本文研究了非结构化网格Galerkin有限元方法对有限域上定义的具有时间无关位势的二维时空分数阶薛定谔方程的误差分析。有限差分法用于离散化Caputo时间分数阶导数，而使用非结构化网格的有限元方法处理空间中的Riesz分数阶算子。建立了数值方案的稳定性和收敛性分析。数值例子验证了所提出方法的有效性。实现了量子力学系统中空间分数阶薛定谔方程的守恒和时间分数阶薛定谔方程的不守恒。