A fast energy conserving finite element method for the nonlinear fractional Schrödinger equation with wave operator

The main aim of this paper is to apply the Galerkin finite element method to numerically solve the nonlinear fractional Schrödinger equation with wave operator. We first construct a fully discrete scheme combining the Crank-Nicolson method with the Galerkin finite element method. Two conserved quantities of the discrete system are shown. Meanwhile, the prior bound of the discrete solutions are proved. Then, we prove that the discrete scheme is unconditionally convergent in the senses of L2−norm and Hα/2−norm. Moreover, by the proposed iterative algorithm, some numerical examples are given to verify the theoretical results and show the effectiveness of the numerical scheme. Finally, a fast Krylov subspace solver with suitable circulant preconditioner is designed to solve above Toeplitz-like linear system. In each iterative step, this method can effectively reduce the memory requirement of the proposed iterative finite element scheme from O(M2) to O(M), and the computational complexity from O(M3) to O(MlogM), where M is the number of grid nodes. Several numerical tests are carried out to show that this fast algorithm is more practical than the traditional backslash and LU factorization methods, in terms of memory requirement and computational cost.