High-order Mass- and Energy-conserving SAV-Gauss Collocation Finite Element Methods for the Nonlinear Schrödinger Equation

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