Linearized Transformed $L1$ Galerkin FEMs with Unconditional Convergence for Nonlinear Time Fractional Schrödinger Equations

Abstract

A linearized transformed $L1$ Galerkin finite element method (FEM) is presented for numerically solving the multi-dimensional time fractional Schrödinger equations. Unconditionally optimal error estimates of the fully-discrete scheme are proved. Such error estimates are obtained by combining a new discrete fractional Grönwall inequality, the corresponding Sobolev embedding theorems and some inverse inequalities. While the previous unconditional convergence results are usually obtained by using the temporal-spatial error spitting approaches. Numerical examples are presented to confirm the theoretical results.