A Novel Two-Grid Finite Element Approach for Solving Nonlinear Time-Fractional Schrödinger Equation

Source abstract

A novel two-grid finite element method with $\mathcal{F}L2-1_\sigma$ scheme is presented for solving nonlinear time-fractional Schrödinger equation. The finite element solution is proven bounded in the $L^\infty$-norm, with no restrictions imposed on the grid ratio. The $\mathcal{F}L2-1_\sigma$ scheme is adopted to construct a second-order numerical scheme in the time direction, while the two-grid finite element method is applied for spatial discretization, which effectively reduces computational storage and cost. The optimal order error estimations of the two-grid solution in the $L^p$-norm are proved without any grid-ratio conditions. Finally, we present some numerical experiments to validate the accuracy and effectiveness of the scheme.