Unconditionally Convergent $L1$-Galerkin FEMs for Nonlinear Time-Fractional Schrödinger Equations

Abstract

In this paper, a linearized $L1$-Galerkin finite element method is proposed to solve the multidimensional nonlinear time-fractional Schrodinger equation. In terms of a temporal-spatial error splitting argument, we prove that the finite element approximations in the $L^2$-norm and $L^\infty$-norm are bounded without any time-step size conditions. More importantly, by using a discrete fractional Gronwall-type inequality, optimal error estimates of the numerical schemes are obtained unconditionally, while the classical analysis for multidimensional nonlinear fractional problems always required certain time-step restrictions dependent on the spatial mesh size. Numerical examples are given to illustrate our theoretical results.