A linearized energy--conservative finite element method for the nonlinear Schr\"{o}dinger equation with wave operator
In this paper, we propose a linearized finite element method (FEM) for solving the cubic nonlinear Schr\"{o}dinger equation with wave operator. In this method, a modified leap-frog scheme is applied for time discretization and a Galerkin finite element method is applied for spatial discretization. We prove that the proposed method keeps the energy conservation in the given discrete norm. Comparing with non-conservative schemes, our algorithm keeps higher stability. Meanwhile, an optimal error estimate for the proposed scheme is given by an error splitting technique. That is, we split the error into two parts, one from temporal discretization and the other from spatial discretization. First, by introducing a time-discrete system, we prove the uniform boundedness for the solution of this time-discrete system in some strong norms and obtain error estimates in temporal direction. With the help of the preliminary temporal estimates, we then prove the pointwise uniform boundedness of the finite element solution, and obtain the optimal $L^2$-norm error estimates in the sense that the time step size is not related to spatial mesh size. Finally, numerical examples are provided to validate the convergence-order, unconditional stability and energy conservation.