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

Khaled Omrani|Mohamed Rahmeni
대한수학회 (The Korean Mathematical Society)|2020.01|대한수학회보 (Bulletin of the KMS)|제57권 제1호|pp. 219-244|DOI 10.4134/BKMS.b190154

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참고문헌

키워드

convergenceconservation lawsCoupled Schr\"{o}dinger equationsGalerkin finite element schemeunique solvability
주제분류수학

초록

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.

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.

참고문헌 (33)

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