SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1566-1591, January 2021.
A family of arbitrarily high-order fully discrete space-time finite element methods are proposed for the nonlinear Schrödinger equation based on the scalar auxiliary variable formulation, which consists of a Gauss collocation temporal discretization and the finite element spatial discretization. The proposed methods are proved to be well-posed and conserving both mass and energy at the discrete level. An error bound of the form $O(h^p+\tau^{k+1})$ in the $L^\infty(0,T;H^1)$-norm is established, where $h$ and $\tau$ denote the spatial and temporal mesh sizes, respectively, and $(p,k)$ is the degree of the space-time finite elements. Numerical experiments are provided to validate the theoretical results on the convergence rates and conservation properties. The effectiveness of the proposed methods in preserving the shape of a soliton wave is also demonstrated by numerical results.

中文翻译：

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非线性薛定谔方程的高阶质量和能量守恒SAV-Gauss搭配有限元方法

SIAM 数值分析杂志，第 59 卷，第 3 期，第 1566-1591 页，2021 年 1 月。
针对非线性薛定谔方程，基于标量辅助变量公式，提出了一系列任意高阶完全离散时空有限元方法，该方法由高斯搭配时间离散化和有限元空间离散化组成。所提出的方法被证明是适定的，并且在离散水平上保持质量和能量守恒。建立$L^\infty(0,T;H^1)$-范数中$O(h^p+\tau^{k+1})$形式的误差界，其中$h$和$ \tau$ 分别表示空间和时间网格大小，$(p,k)$ 是时空有限元的度数。提供了数值实验来验证收敛速度和守恒性质的理论结果。