On Fully Discrete Galerkin Methods of Second–order Temporal Accuracy for the Nonlinear Schrödinger Equation

We approximate the solutions of an initialand boundary-value problem for nonlinear Schrödinger equations (with emphasis on the ‘cubic’ nonlinearity) by two fully discrete finite element schemes based on the standard Galerkin method in space and two implicit, Crank–Nicolson-type second-order accurate temporal discretizations. For both schemes we study the existence and uniqueness of their solutions and prove L error bounds of optimal order of accuracy. For one of the schemes we also analyze one step of Newton’s method for solving the nonlinear systems that arise at every time step. We then implement this scheme using an iterative modification of Newton’s method that, at each time step tn, requires solving a number of sparse complex linear systems with a matrix that does not change with n. The effect of this ‘inner’ iteration is studied theoretically and numerically.