An efficient finite element method for the two-dimensional nonlinear time-space fractional Schrödinger equation on an irregular convex domain

In this paper, we introduce an efficient Galerkin finite element method for solving the two-dimensional nonlinear time-space fractional Schrödinger equation, with time-dependent potential function, defined on an irregular convex domain. The Caputo time fractional derivative is discretized by the finite difference scheme, while the space fractional derivative defined on the irregular convex domain is dealt with the unstructured mesh finite element method. Given the nonlinear property of the considered equation, an efficient linearized strategy is used. To testify the efficiency of the proposed numerical scheme, two numerical examples are conducted with error and convergence analysis.