Exponential decay estimates of a general linear weakly damped wave equation are studied with decay rate lying in a range. Based on the $C^0$-conforming finite element method to discretize spatial variables keeping temporal variable continuous, a semidiscrete system is analysed, and uniform decay estimates are derived with precisely the same decay rate as in the continuous case. Optimal error estimates with minimal smoothness assumptions on the initial data are established, which preserve exponential decay rate, and for a 2D problem, the maximum error bound is also proved. The present analysis is then generalized to include the problems with non-homogeneous forcing function, space-dependent damping, and problems with compensator. It is observed that decay rates are improved with large viscous damping and compensator. Finally, some numerical experiments are performed to validate the theoretical results established in this paper.

翻译：对一般线性微弱倾斜波方程式的指数衰变估计值进行研究,研究的衰变率范围为零。根据将空间变量分解以保持时间变量连续性的$C$0的兼容的有限要素法,对半分解系统进行了分析,并用与连续情况完全相同的衰变率得出统一的衰变估计值。在初步数据上设定了最佳误差估计值,以最小平滑的假设维持指数衰变率,对于2D问题,最大误差约束也得到了证明。本分析随后被广泛推广,包括非同源强制功能的问题、空间依赖的倾斜以及同化器的问题。观察到,与大型粘粘土堆和计算器相比,衰变率得到了改善。最后,进行了一些数字实验,以验证本文确定的理论结果。</s>