This paper discusses the initial-boundary value problem for the cubic Schrödinger equation subject to and on the boundary.
The approximate solutions are obtained using Galerkin finite element methods and implicit Runge-Kutta schemes for the time dependence. An error bound is derived and the temporal component of the discretization error is shown to decrease at classical rates in some important special cases.
The strategy of the proof of the error bound is clearly explained. This includes constructing a smooth approximation at intermediate times of the Runge-Kutta scheme and dealing with the problem of ensuring that it vanishes on the boundary.