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In this paper, we propose a linearized finite element method (FEM) forsolving the cubic nonlinear Schr\"{o}dinger equation with wave operator. Inthis method, a modified leap-frog scheme is applied for time discretization anda Galerkin finite element method is applied for spatial discretization. Weprove that the proposed method keeps the energy conservation in the givendiscrete norm. Comparing with non-conservative schemes, our algorithm keepshigher stability. Meanwhile, an optimal error estimate for the proposed schemeis given by an error splitting technique. That is, we split the error into twoparts, one from temporal discretization and the other from spatialdiscretization. First, by introducing a time-discrete system, we prove theuniform boundedness for the solution of this time-discrete system in somestrong norms and obtain error estimates in temporal direction. With the help ofthe preliminary temporal estimates, we then prove the pointwise uniformboundedness of the finite element solution, and obtain the optimal $L^2$-normerror estimates in the sense that the time step size is not related to spatialmesh size. Finally, numerical examples are provided to validate theconvergence-order, unconditional stability and energy conservation.

A linearized energy--conservative finite element method for the nonlinear Schr\"{o}dinger equation with wave operator