Unconditionally convergent L1-Galerkin FEMs for nonlinear time-fractional SchrÖdinger equations

In this paper, a linearized L1-Galerkin finite element method is proposed to solve the multidimensional nonlinear time-fractional Schrödinger equation. In terms of a temporal-spatial error splitting argument, we prove that the finite element approximations in the L²-norm and L^∞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.