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

TL;DR

This paper introduces an energy-conserving Galerkin finite element method using Crank-Nicolson time discretization for generalized Klein-Gordon-Zakharov equations, with rigorous error analysis and numerical validation.

Contribution

It develops a novel energy-conservation FEM with superconvergence properties and provides comprehensive error bounds for key variables in the equations.

Findings

Exact energy conservation in the discrete scheme.

Superclose error estimates and superconvergence results.

Numerical examples confirm theoretical error bounds.

Abstract

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.