Optimal error estimates of conservative virtual element method for the coupled nonlinear Schrödinger–Helmholtz equation

Abstract

In this work, we propose a novel class of mass- and energy-conserving schemes formulated on arbitrary polygonal meshes for the coupled nonlinear Schrödinger–Helmholtz system. This approach leverages the Crank–Nicolson time discretization and the virtual element method for spatial discretization. To establish the theoretical foundation, we use the duality argument to estimate the difference quotient of the error in the $H^{-1}\left(\Omega \right)$-norm and the classical Schaefer’s fixed point theorem to demonstrate the existence, uniqueness, and convergence of the numerical solutions when $h$ and $\tau$ are sufficiently small. Specifically, we rigorously derive an optimal error estimate of the form $O({\tau }^2+h^r)$ in the $L^{\infty }(0,T;H^1)$-norm without restriction on the grid ratio, where $\tau$ and $h$ represent the temporal and spatial mesh sizes, respectively, and $r$ is the degree of approximation. Compared to conventional theoretical analysis techniques, our methodology does not require temporal–spatial splitting arguments and avoids cumbersome mathematical induction. Finally, numerical examples on a set of polygonal meshes confirm the accuracy and efficacy of our proposed method, underscoring its conservation properties over long-time simulations.