Space–Time Methods Based on Isogeometric Analysis for Time-fractional Schrödinger Equation

In this paper, we propose a time discontinuous Galerkin scheme for solving the nonlinear time-fractional Schrödinger equation using B-splines in time and Non-Uniform Rational B-splines in space. The technique of comparing real and imaginary parts is utilized to obtain optimal L 2 ( [ 0 , T ] ; L 2 ( Ω ) ) norm error estimate. Specifically, we have achieved r + 1 accuracy in time and p + 1 accuracy in space, where r and p represent the spline degrees in time and space, respectively. The convergence analysis is also provided on time graded mesh, taking into account solutions with initial singularity. Additionally, the space–time isogeometric analysis method is employed to solve the linear time-fractional Schrödinger equation. A new discrete norm is constructed, and the well-posedness analysis and error estimate are performed based on this norm. We can attain p ^ accuracy concerning the new discrete norm error in space–time domain, where p ^ denotes space–time spline degree. Theoretical results are validated through using numerical examples.