A novel energy-conservation Crank-Nicolson finite element method for generalized Klein-Gordon-Zakharov equations
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866916004119969792 |
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| author | Xu, Xuemiao Jiang, Maosheng Zhang, Jiansong Zhu, Jiang |
| author_facet | Xu, Xuemiao Jiang, Maosheng Zhang, Jiansong Zhu, Jiang |
| contents | 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 $φ$. 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 $ϕ$ ($-Δϕ= φ_t$) and optimal-order $L^2$-norm error estimates for the auxiliary variable $p$ ($p=u_t$) and $φ$. Numerical examples are provided to confirm theoretical results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_11686 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A novel energy-conservation Crank-Nicolson finite element method for generalized Klein-Gordon-Zakharov equations Xu, Xuemiao Jiang, Maosheng Zhang, Jiansong Zhu, Jiang Numerical Analysis 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 $φ$. 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 $ϕ$ ($-Δϕ= φ_t$) and optimal-order $L^2$-norm error estimates for the auxiliary variable $p$ ($p=u_t$) and $φ$. Numerical examples are provided to confirm theoretical results. |
| title | A novel energy-conservation Crank-Nicolson finite element method for generalized Klein-Gordon-Zakharov equations |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2605.11686 |