Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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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.

Error estimates for a Galerkin method for a coupled nonlinear Schr\"{o}dinger equations

Khaled Omrani, Mohamed Rahmeni

Universit\'e de Sousse; Universit\'e de Sousse

Abstract

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