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：研究了一般线性弱阻尼波动方程在衰减率处于某个区间内的指数衰减估计。基于 $C^0$-协调有限元方法对空间变量进行离散，同时保持时间变量连续，分析了半离散系统，并推导出与连续情况完全相同的均匀衰减率。在初始数据的最小光滑性假设下，建立了保持指数衰减率的最优误差估计；对于二维问题，还证明了最大误差界。随后将现有分析推广至包含非齐次强迫函数、空间相关阻尼以及带补偿器的问题。研究发现，大粘性阻尼和补偿器可改善衰减率。最后，通过数值实验验证了本文建立的理论结果。