(untitled) (Journal of Computational Mathematics, 2024)
In this work, we focus on the conforming and nonconforming leap-frog virtual element methods for the generalized nonlinear Schrödinger equation, and establish their unconditional stability and optimal error estimates. By constructing a time-discrete system, the error between the solutions of the continuous model and the numerical scheme is separated into the temporal error and the spatial error, which makes the spatial error τ-independent. The inverse inequalities in the existing conforming and new constructed nonconforming virtual element spaces are utilized to derive the L^∞-norm uniform boundedness of numerical solutions without any restrictions on time-space step ratio, and then unconditionally optimal error estimates of the numerical schemes are obtained naturally. What needs to be emphasized is that if we use the pre-existing nonconforming virtual elements, there is no way to derive the L^∞-norm uniform boundedness of the functions in the nonconforming virtual element spaces so as to be hard to get the corresponding inverse inequalities. Finally, several numerical examples are reported to confirm our theoretical results.