We prove optimal H1 estimates for the backward Euler and Crank-Nicolson discretizations of the Galerkin finite element method applied to a nonlinear Schrödinger equation. As a by-product, we also obtain new pointwise error bounds. The analysis relies on a nonlinear stability theory recently developed by López-Marcos & Sanz-Serna 1991.
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Yves Tourigny (1991) studied this question.