Finite Element Methods for Variational Integrators with Applications to Nonlinear Schrödinger Equation

In this paper we introduce a new method of spatio-temporal discretization for partial differential equations in variational form. This method generalizes the method of Marsden et al in that it uses a systematic approach to discrete jet spaces based on the finite element method. The resulting method is used to derive integrators for the Nonlinear Schrödinger (NLS) equation which exhibit superior conservation properties. In particular, the second order variational integrators that we derive conserve the three leading integrals of the motion of the linear and NLS equations several orders of magnitude more than is dictated by the discretization order. Furthermore, the variational integrators are superior even to the second order implicit RungeKutta method, which is symplectic for systems with a canonical Poisson structure. Finally, the use of the finite element method suggests a systematic approach to the convergence analysis of the integrators.