| Abstract: |
| In this paper, a linearized $L1$-Galerkin finite element method is proposed to solve the multi-dimensional nonlinear time-fractional Schr\{o}dinger equation. In terms of a temporal-spatial error splitting argument,
we prove that the finite element approximations in $L^2$-norm and $L^\infty$-norm are bounded without any time stepsize 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 multi-dimensional nonlinear fractional problems always required certain time-step restrictions dependent on the spatial mesh size. Numerical examples are given to illustrate our theoretical results. |
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