Finite element method for time-space-fractional Schrodinger equation

Abstract

In this article, we develop a fully discrete finite element method for the nonlinear Schrodinger equation (NLS) with time- and space-fractional derivatives. The time-fractional derivative is described in Caputo's sense and the space-fractional derivative in Riesz's sense. Its stability is well derived; the convergent estimate is discussed by an orthogonal operator. We also extend the method to the two-dimensional time-space-fractional NLS and to avoid the iterative solvers at each time step, a linearized scheme is further conducted. Several numerical examples are implemented finally, which confirm the theoretical results as well as illustrate the accuracy of our methods.

Keywords

• Time-space-fractional NLS
• finite element method
• convergence