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Galerkin finite element method for nonlinear fractional Schrödinger equations. (English) Zbl 1359.65208

The authors use the Crank-Nicolson scheme in time and the finite element method in space to solve a class of nonlinear Riesz space-fractional Schrödinger equations. Using a Brouwer fixed point theorem and a fractional Gagliardo-Nirenberg inequality, unique solvability of fully discrete systems is proved. The conservation and convergence properties of the semi-discrete scheme and the fully discrete one are analyzed. A linearized iterative finite element algorithm is introduced and some numerical examples are given to support the theoretical results.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs

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