In this paper, an energy-based discontinuous Galerkin method for dynamic Euler-Bernoulli beam equations is developed. The resulting method is energy-dissipating or energy-conserving depending on the simple, mesh-independent choice of numerical fluxes. By introducing a velocity field, the original problem is transformed into a first-order in time system. In our formulation, the discontinuous Galerkin approximations for the original displacement field and the auxiliary velocity field are not restricted to be in the same space. In particular, a given accuracy can be achieved with the fewest degrees of freedom when the degree for the approximation space of the velocity field is two orders lower than the degree of approximation space for the displacement field. In addition, we establish the error estimates in an energy norm and demonstrate the corresponding optimal convergence in numerical experiments.

中文翻译：

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动态欧拉-伯努利梁方程的一种基于能量的不连续伽辽金方法

在本文中，开发了一种用于动态 Euler-Bernoulli 梁方程的基于能量的不连续 Galerkin 方法。由此产生的方法是耗能的还是能量守恒的，这取决于简单的、与网格无关的数值通量选择。通过引入速度场，原问题转化为一阶时间系统。在我们的公式中，原始位移场和辅助速度场的不连续伽辽金近似不限于在同一空间中。特别是，当速度场的近似空间的度比位移场的近似空间的度低两个数量级时，可以用最少的自由度实现给定的精度。此外，