The main aim of this paper is to propose a Galerkin finite element method (FEM) for solving the Klein–Gordon–Zakharov (KGZ) equations with power law nonlinearity, and to give the error estimations of approximate solutions about the electronic fast time scale component $q$ and the ion density deviation $r$. In which, the bilinear element is used for spatial discretization, and a second order difference scheme is implemented for temporal discretization. Moreover, by use of the combination of the interpolation and Ritz projection technique, and the interpolated postprocessing approach, for weaker regularity requirements of the exact solution, the superclose and global superconvergence estimations of $q$ in ${\mathcal{H}}^1-$norm are deduced. At the same time, the superconvergence of the auxiliary variable $\phi$ ($-\Delta \phi =r_t$) in $H^1-$norm and optimal error estimation of $r$ in $L^2-$norm are derived. Meanwhile, we also discuss the extensions of our scheme to more general finite elements. Finally, the numerical experiments are provided to confirm the validity of the theoretical analysis.

本文的主要目的是提出一种用于求解具有幂律非线性的 Klein-Gordon-Zakharov (KGZ) 方程的 Galerkin 有限元方法 (FEM)，并给出关于电子快速时标分量的近似解的误差估计 $q$ 和离子密度偏差 $r$. 其中，双线性元用于空间离散化，时间离散化采用二阶差分格式。此外，通过使用插值和 Ritz 投影技术的结合，以及插值后处理方法，对于精确解的较弱正则性要求，超接近和全局超收敛估计$q$ 在 ${\mathcal{H}}^1-$推导出范数。同时，辅助变量的超收敛$\phi$ ($-\Delta \phi =r_{吨}$） 在 $H^1-$范数和最优误差估计 $r$ 在 ${升}^2-$范数是推导出来的。同时，我们还讨论了我们的方案对更一般的有限元的扩展。最后,通过数值实验验证了理论分析的有效性。