In this paper, a linearized decoupled mass and energy conservation Crank-Nicolson (CN) fully-discrete scheme is proposed for the coupled nonlinear Schrödinger (CNLS) system with the conforming bilinear Galerkin finite element method (FEM), and the unconditional supercloseness and superconvergence error estimates in \(H^1\) -norm are deduced rigorously. Firstly, with the aid of the popular time-space splitting technique, that is, by introducing a suitable time discrete system, the error is divided into two parts, the time error and spatial error, the boundedness of numerical solution in \(L^\infty \) -norm is derived strictly without any constraint between the mesh size h and the time step \(\tau \) . Then, thanks to the high accuracy result between the interpolation and Ritz projection, the unconditional superclose error estimate is obtained, and the corresponding unconditional superconvergence result is acquired through the interpolation post-processing technique. At last, some numerical results are supplied to verify the theoretical analysis.
