Physics

Finite element method

Nonlinear system

Temporal discretization

Mathematics

Schrödinger equation

Nonlinear Schrödinger equation

Mathematical analysis

Political science

Norm (philosophy)

Gauss

Geometry

Numerical analysis

Quantum mechanics

Discretization

Applied mathematics

Scalar (mathematics)

Thermodynamics

SIAM Journal on Numerical Analysis

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.

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