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Galerkin finite element method for nonlinear fractional Schrödinger equations

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Abstract

In this paper, a class of nonlinear Riesz space-fractional Schrödinger equations are considered. Based on the standard Galerkin finite element method in space and Crank-Nicolson difference method in time, the semi-discrete and fully discrete systems are constructed. By Brouwer fixed point theorem and fractional Gagliardo-Nirenberg inequality, we prove the fully discrete system is uniquely solvable. Moreover, we focus on a rigorous analysis and consideration of the conservation and convergence properties for the semi-discrete and fully discrete systems. Finally, a linearized iterative finite element algorithm is introduced and some numerical examples are given to confirm the theoretical results.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China

    Meng Li, Chengming Huang & Pengde Wang

Authors
  1. Meng Li
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  2. Chengming Huang
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  3. Pengde Wang
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Corresponding author

Correspondence to Chengming Huang.

Additional information

This work was supported by NSF of China (No. 11371157) and NSF of Anhui Higher Education Institutions of China (No. KJ2016A492).

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Li, M., Huang, C. & Wang, P. Galerkin finite element method for nonlinear fractional Schrödinger equations. Numer Algor 74, 499–525 (2017). https://doi.org/10.1007/s11075-016-0160-5

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  • DOI: https://doi.org/10.1007/s11075-016-0160-5

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