A fast linearized conservative finite element method for the strongly coupled nonlinear fractional Schrödinger equations

• Published in: Journal of computational physics Vol. 358; pp. 256 - 282
• Main Authors: Li, Meng, Gu, Xian-Ming, Huang, Chengming, Fei, Mingfa, Zhang, Guoyu
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 01.04.2018; Elsevier Science Ltd
• Subjects: Circulant preconditioner; Computational mathematics; Computational physics; Computer simulation; Coupled nonlinear fractional Schrödinger equations; Energy conservation; Finite element analysis; Finite element method; Iterative algorithms; Iterative methods; Krylov subspace methods; Linear systems; Linearization; Mass and energy conservation; Mathematical analysis; Matrix methods; Nonlinear equations; Nonlinear systems; Numerical analysis; Schrodinger equation; Solitary waves; Solvers; Unconditional convergence; Finite element method; Coupled nonlinear fractional Schrödinger equations; Mass and energy conservation; Krylov subspace methods; Circulant preconditioner; Unconditional convergence
• ISSN: 0021-9991; 1090-2716; 1090-2716
• DOI: 10.1016/j.jcp.2017.12.044

In this paper, a fast linearized conservative finite element method is studied for solving the strongly coupled nonlinear fractional Schrödinger equations. We prove that the scheme preserves both the mass and energy, which are defined by virtue of some recursion relationships. Using the Sobolev inequalities and then employing the mathematical induction, the discrete scheme is proved to be unconditionally convergent in the sense of L2-norm and Hα/2-norm, which means that there are no any constraints on the grid ratios. Then, the prior bound of the discrete solution in L2-norm and L∞-norm are also obtained. Moreover, we propose an iterative algorithm, by which the coefficient matrix is independent of the time level, and thus it leads to Toeplitz-like linear systems that can be efficiently solved by Krylov subspace solvers with circulant preconditioners. This method can reduce the memory requirement of the proposed linearized finite element scheme from O(M2) to O(M) and the computational complexity from O(M3) to O(Mlog⁡M) in each iterative step, where M is the number of grid nodes. Finally, numerical results are carried out to verify the correction of the theoretical analysis, simulate the collision of two solitary waves, and show the utility of the fast numerical solution techniques. •A new linearized conservative FEM for the strongly coupled nonlinear fractional Schrödinger equations is proposed.•The mass and energy conservation results are obtained for the proposed discretized scheme.•The scheme is proved to be unconditionally convergent in both the L2-norm and fractional norm.•To reduce the memory requirement, a super-fast method is used for the iterative algorithm of the proposed linearized FEM.