Galerkin finite element method for the nonlinear fractional Ginzburg-Landau equation

In this paper, we are concerned with the numerical solution of the nonlinear fractional Ginzburg-Landau equation. Galerkin finite element method is used for the spatial discretization, and an implicit midpoint difference method is employed for the temporal discretization. The boundedness, existence and uniqueness of the numerical solution, and the unconditional error estimates in the L2-norm are investigated in details. To numerically solve the nonlinear system, linearized iterative algorithms are also considered. Finally, some numerical examples are presented to illustrate the effectiveness of the algorithm.