Optimal Error Estimates of SAV Crank–Nicolson Finite Element Method for the Coupled Nonlinear Schrödinger Equation

In this paper, we reformulate the coupled nonlinear Schrödinger (CNLS) equation by using the scalar auxiliary variable (SAV) approach and solve the resulting system by using Crank-Nicolson finite element method. The fully-discrete method is proved to be mass- and energy-conserved. However, if the convergence results are investigated by using the classical way, the presence of u_t and v_t in the equation of r'(t) may lead to not only a consistency error of sub-optimal order in time but also some difficulties in analysing the numerical stability. The mentioned difficulties are overcome technically by estimating the difference quotient of the error in the H^-1 -norm and carefully analysising the connections of errors between the couple systems. Consequently, the numerical solution is shown to be convergent at the order of 𝒪( τ ^2 + h^p) in the H^1 -norm with time step τ , mesh size h and the degree of finite elements p . Several numerical examples are presented to confirm our theoretical results.