Abstract In this paper for the first time, two kinds of energy-preserving finite element approximation schemes, which are based upon the standard finite element method (FEM) and the mixed FEM, respectively, are developed and analyzed for a class of nonlinear wave equations. The energy conservation and the optimal convergence properties are obtained for both finite element schemes in their respective norms, additionally, the energy-preserving mixed FEM can produce one-order higher approximation accuracy to the flux (the gradient of the primary unknown) in L 2 norm in contrast with that of the standard FEM when the same degree piecewise polynomial is employed to construct their respective finite element spaces, which may likely result in a more accurate and more physical discrete energy conservation. Numerical experiments are carried out to validate all attained theoretical results. Furthermore, the developed energy-preserving finite element methods can be directly applied to the coupled system of nonlinear wave equations, whose energy conservation and optimal convergence properties are also confirmed by our numerical experiments.

中文翻译：

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一类非线性波动方程的保能有限元方法

摘要 本文首次针对一类非线性波动方程，分别基于标准有限元法(FEM)和混合有限元法(FEM)开发并分析了两种保能有限元近似方案。 . 两种有限元方案都在各自的范数下获得了能量守恒和最优收敛特性，此外，能量保持混合有限元法可以对 L 2 中的通量（主要未知数的梯度）产生一阶更高的近似精度范数与标准 FEM 的范数相反，当使用相同的阶数分段多项式来构造它们各自的有限元空间时，这可能会导致更准确和更物理的离散能量守恒。进行数值实验以验证所有获得的理论结果。此外，所开发的能量守恒有限元方法可以直接应用于非线性波动方程的耦合系统，其能量守恒和最优收敛性也得到了我们的数值实验的证实。