A novel energy-conservation Crank-Nicolson finite element method for generalized Klein-Gordon-Zakharov equations

This article focuses on an energy-conservation Galerkin finite element method (FEM) for the generalized Klein-Gordon-Zakharov (KGZ) equations. This method combines the bilinear finite element method for spatial discretization with the Crank-Nicolson (CN) scheme for temporal discretization, thereby guaranteeing exact conservation of the discrete energy functional. A rigorous theoretical analysis is devoted to deriving error bounds for the fast-time-scale electronic field $u$ and the ion density deviation $\varphi$. By systematically integrating interpolation estimates, Ritz projection, and a postprocessing technique, the superclose error estimates and global superconvergence are established for $u$ in the $H^1$-norm, even under weakened regularity assumptions on the exact solution. Concurrently, we prove $H^1$-norm superconvergence for the auxiliary variable $\phi$ ($-\Delta\phi = \varphi_t$) and optimal-order $L^2$-norm error estimates for the auxiliary variable $p$ ($p=u_t$) and $\varphi$. Numerical examples are provided to confirm theoretical results.