This paper proposes an energy-based discontinuous Galerkin scheme for fourth-order semilinear wave equations, which we rewrite as a system of second-order spatial derivatives. Compared to the local discontinuous Galerkin methods, the proposed scheme uses fewer auxiliary variables and is more computationally efficient. We prove several properties of the scheme. For example, we show that the scheme is unconditionally stable and that it achieves optimal convergence in $L^2$ norm for both the solution and the auxiliary variables without imposing penalty terms. A key part of the proof of the stability and convergence analysis is the special choice of the test function for the auxiliary equation involving the time derivative of the displacement variable, which leads to a linear system for the time evolution of the unknowns. Then we can use standard mathematical techniques in discontinuous Galerkin methods to obtain stability and optimal error estimates. We also obtain energy dissipation and/or conservation of the scheme by choosing simple and mesh-independent interelement fluxes. Several numerical experiments are presented to illustrate and support our theoretical results.

中文翻译：

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基于局部能量的四阶半线性波动方程间断伽辽金法

本文提出了一种基于能量的四阶半线性波动方程的不连续伽辽金格式，我们将其重写为二阶空间导数系统。与局部不连续伽辽金方法相比，该方案使用的辅助变量更少，计算效率更高。我们证明了该方案的几个属性。例如，我们证明该方案是无条件稳定的，并且它在解和辅助变量的 $L^2$ 范数中实现了最优收敛，而无需施加惩罚项。稳定性和收敛性分析证明的一个关键部分是对涉及位移变量的时间导数的辅助方程的检验函数的特殊选择，这导致了未知数时间演化的线性系统。然后我们可以在不连续伽辽金方法中使用标准数学技术来获得稳定性和最优误差估计。我们还通过选择简单且与网格无关的元素间通量来获得该方案的能量耗散和/或守恒。提出了几个数值实验来说明和支持我们的理论结果。