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On optimal order error estimates for the nonlinear Schrödinger equation. (English) Zbl 0774.65091
This paper discusses the initial-boundary value problem for the cubic Schrödinger equation $$u_ t=i\Delta u+i\lambda| u|^ 2 u$$ subject to $$u(x,0)=u^ 0(x)$$ and $$u=0$$ 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.

##### MSC:
 65Z05 Applications to the sciences 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M20 Method of lines for initial value and initial-boundary value problems involving PDEs 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 35Q55 NLS equations (nonlinear Schrödinger equations) 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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