Two-grid finite element methods for space-fractional nonlinear Schrödinger equations

A two-grid finite element method is developed for solving space-fractional nonlinear Schrödinger equations. The finite element solution in $L^{\infty }$-norm is proved bounded without any time-step size conditions (dependent on spatial-step size). Then, the optimal order error estimations of the two-grid solution in the $L^p$-norm are proved without any time-step size conditions. Finally, the theoretical results are verified by numerical experiments.