In this paper we consider a mass- and energy--conserving Crank-Nicolson time discretization for a general class of nonlinear Schrodinger equations. This scheme, which enjoys popularity in the physics community due to its conservation properties, was already subject to several analytical and numerical studies. However, a proof of optimal $L^{\infty}(H^1)$-error estimates is still open, both in the semi-discrete Hilbert space setting, as well as in fully-discrete finite element settings. This paper aims at closing this gap in the literature.

中文翻译：

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关于非线性薛定谔方程的 Crank-Nicolson 近似的最优 $$H^1$$-误差估计的说明

在本文中，我们考虑对一类一般非线性薛定谔方程进行质量和能量守恒的 Crank-Nicolson 时间离散化。由于其守恒特性，该方案在物理学界广受欢迎，已经进行了多项分析和数值研究。然而，最佳 $L^{\infty}(H^1)$ 误差估计的证明仍然是开放的，无论是在半离散希尔伯特空间设置中，还是在完全离散有限元设置中。本文旨在缩小文献中的这一差距。