Abstract: Energy conservation of nonlinear Schrödinger ordinary differential equation was proved through using continuous finite element methods of ordinary differential equation; Energy integration conservation was proved through using space-time continuous fully discrete finite element methods and the electron nearly conservation with higher order error was obtained through using time discontinuous only space continuous finite element methods of nonlinear Schrödinger partial equation. The numerical results are in accordance with the theory.

Key words: energy integration, nonlinear Schrö, dinger equation, space-time finite element method, conservation