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High accuracy analysis of Galerkin finite element method for Klein–Gordon–Zakharov equations

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  • Shi, Dongyang
  • Wang, Ran

Abstract

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 H1−norm are deduced. At the same time, the superconvergence of the auxiliary variable φ (−Δφ=rt) in H1−norm and optimal error estimation of r in L2−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.

Suggested Citation

  • Shi, Dongyang & Wang, Ran, 2022. "High accuracy analysis of Galerkin finite element method for Klein–Gordon–Zakharov equations," Applied Mathematics and Computation, Elsevier, vol. 415(C).
    • Handle: RePEc:eee:apmaco:v:415:y:2022:i:c:s0096300321007852
    DOI: 10.1016/j.amc.2021.126701
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    1. Thoudam, Roshan, 2017. "Numerical solutions of coupled Klein–Gordon–Zakharov equations by quintic B-spline differential quadrature method," Applied Mathematics and Computation, Elsevier, vol. 307(C), pages 50-61.
    2. Zhang, Houchao & Shi, Dongyang & Li, Qingfu, 2020. "Nonconforming finite element method for a generalized nonlinear Schrödinger equation," Applied Mathematics and Computation, Elsevier, vol. 377(C).
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