Energy conservation of non-linear Schrêdinger ordinary differential equation was proved through using ordinary differential equation's continuous finite element methods;Energy integration conservation was proved through using space-time all continuous fully discrete finite element methods and electron nearly conservation with higher order error through using time discontinuous only space continuos finite element methods of non-linear Schrêdinger partial equation.The numerical results are in accordance with the theory.
Delfour M,Fortin M,Payre G. Finite-difference solution of a non-linear Schrdinger equation[J].Journal of Computational Physics,1981,44(12):277—288. doi: 10.1016/0021-9991(81)90052-8
Qhannes Karakashian,Charalambos Makridakis.A space-time finite element method for The nonlinear Schrdinger equation:the discontiouous Galerkin method[J].Mathematics of Computation,1998,67(1):479—499. doi: 10.1090/S0025-5718-98-00946-6