High-order mass- and energy-conserving SAV-Gauss collocation finite element methods for the nonlinear Schr\"odinger equation

A family of arbitrarily high-order fully discrete space-time finite element methods are proposed for the nonlinear Schr\"odinger 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.