Bull. Korean Math. Soc. 57 (2020), No. 1, pp. 219–244 https://doi.org/10.4134/BKMS.b190154 pISSN: 1015-8634 / eISSN: 2234-3016 ERROR ESTIMATES FOR A GALERKIN METHOD FOR A COUPLED NONLINEAR SCHR ¨ ODINGER EQUATIONS Khaled Omrani and Mohamed Rahmeni 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 mid- point discretization in time. The proposed scheme guarantees the con- servation 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, con- vergence orders of the fully discrete Crank-Nicolson scheme are discussed. The end of the paper is devoted to some numerical experiments. 1. Introduction The time-dependent Nonlinear Schr¨ odinger equations are one of the most important mathematical models. These equations models many physical phe- nomena such as quantum mechanics, optics, seismology, bimolecular dynamics. For more details, we refer the reader to [4,10,11,20,23–25,30–33] and references therein. Recently, a growing interest is focused on the numerical methods (of finite difference, finite element, spectral or more specialized type) for coupled nonlinear Schr¨ odinger (CNLS) equations and Schr¨ odinger equation. See, for example, [3, 8, 9, 13, 14, 16, 17, 19, 21, 26–29] and their lists of references. The main purpose of this paper is analyze a fully discrete Crank-Nicolson type Galerkin method for the following coupled nonlinear Schr¨ odinger (CNLS) equations: (1) iu t + ku xx +(|u| 2 + β|v| 2 )u =0, x ∈ Ω, 0 <t ≤ T, (2) iv t + kv xx +(|v| 2 + β|u| 2 )v =0, x ∈ Ω, 0 <t ≤ T, (3) u(x, 0) = u 0 (x),v(x, 0) = v 0 (x),x ∈ ¯ Ω, Received February 8, 2019; Revised May 30, 2019; Accepted June 26, 2019. 2010 Mathematics Subject Classification. 65M06, 65M12, 65M15. Key words and phrases. Coupled Schr¨ odinger equations, Galerkin finite element scheme, conservation laws, unique solvability, convergence. c 2020 Korean Mathematical Society 219