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 useful for many applications for some instances the former assumption is too strong. Thus in the present article we report a weaker criterion 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：本文研究弱T-强制算子的近似问题。确保其收敛性的主要条件是近似格式的正则性（基于离散逼近方案的术语体系）。前期研究已证明，若存在离散算子$T_n$在离散范数意义下收敛于$T$，则足以保证正则性。虽然该框架已在众多应用场景中验证有效，但对于某些情形，前述假设条件过强。因此，本文提出一种更弱的判别准则：仅需离散算子$T_n$满足逐点收敛，但须在离散层面上附加弱T-强制条件。我们将新框架应用于特定$H^1$-协调有限元离散格式的收敛性证明，该格式用于离散化阻尼时谐Galbrun方程——该方程常用于恒星振荡建模。后续分析的核心要素是散度算子在某些空间上的一致稳定可逆性，此性质与Stokes方程无散元构造理论密切相关。