Bull. Korean Math. Soc. 2020; 57(1): 219-244
Online first article August 2, 2019 Printed January 31, 2020
https://doi.org/10.4134/BKMS.b190154
Copyright © The Korean Mathematical Society.
Khaled Omrani, Mohamed Rahmeni
Universit\'e de Sousse; Universit\'e de Sousse
In this paper, we approximate the solution of the coupled nonlinear Schr\"odinger equations by using a fully discrete finite element scheme based on the standard Galerkin method in space and implicit midpoint discretization in time. The proposed scheme guarantees the conservation of the total mass and the energy. First, a priori error estimates for the fully discrete Galerkin method is derived. Second, the existence of the approximated solution is proved by virtue of the Brouwer fixed point theorem. Moreover, the uniqueness of the solution is shown. Finally, convergence orders of the fully discrete Crank-Nicolson scheme are discussed. The end of the paper is devoted to some numerical experiments.
Keywords: Coupled Schr\"{o}dinger equations, Galerkin finite element scheme, conservation laws, unique solvability, convergence
MSC numbers: 65M06, 65M12, 65M15
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