We consider the approximation of weakly T-coercive operators. The main property to ensure the convergence thereof is the regularity of the approximation (in the vocabulary of discrete approximation schemes). In a previous work the existence of discrete operators $T_n$ which converge to $T$ in a discrete norm was shown to be sufficient to obtain regularity. Although this framework proved usefull for many applications for some instances the former assumption is too strong. Thus in the present article we report a weaker criterium for which the discrete operators $T_n$ only have to converge point-wise, but in addition a weak T-coercivity condition has to be satisfied on the discrete level. We apply the new framework to prove the convergence of certain $H^1$-conforming finite element discretizations of the damped time-harmonic Galbrun's equation, which is used to model the oscillations of stars. A main ingredient in the latter analysis is the uniformly stable invertibility of the divergence operator on certain spaces, which is related to the topic of divergence free elements for the Stokes equation.

翻译：摘要：本文考虑弱T-强制算子的逼近问题。确保其收敛性的主要性质是逼近的正则性（采用离散逼近方案中的术语）。在先前的工作中，已证明存在离散算子$T_n$在离散范数下收敛于$T$足以保证正则性。尽管该框架在许多应用中有效，但在某些情形下上述假设过于严格。因此，本文提出一个更宽松的准则：仅要求离散算子$T_n$逐点收敛，但需在离散层面满足弱T-强制条件。我们应用这一新框架证明了阻尼时谐Galbrun方程（用于模拟恒星振荡）的某些$H^1$相容有限元离散格式的收敛性。分析中的关键要素是散度算子在特定空间上的一致可逆稳定性，该性质与Stokes方程的无散度元问题密切相关。