In this note we simplify the derivation of the error estimates for the generalized Babuška–Brezzi theory with Galerkin schemes defined in terms of approximate bilinear forms and functionals. More precisely, we provide a straight proof that makes no use of any translated continuous or discrete kernel nor of the distance between them, but of suitable upper bounds of the distances of each component of the Galerkin solution to any other member of the respective finite element subspace. In this way, the Strang error estimates are obtained simply by applying the aforementioned bounds along with the triangle inequality, so that they become cleaner and with fully explicit constants. The case in which the discrete bilinear forms can be evaluated at the continuous solution is also considered, which yields the consistency terms to appear separately from the distances to the subspaces, thus allowing the former to be handled independently from the latter.

中文翻译：

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关于广义 Babuška-Brezzi 理论的注释：重新审视相关斯特朗误差估计的证明


在本文中，我们使用根据近似双线性形式和泛函定义的伽辽金方案简化了广义 Babuška-Brezzi 理论的误差估计的推导。更准确地说，我们提供了一个直接证明，该证明不使用任何平移的连续或离散核，也不使用它们之间的距离，而是使用伽辽金解的每个分量到相应有限元的任何其他成员的距离的合适上限子空间。通过这种方式，只需应用上述界限以及三角不等式即可获得斯特朗误差估计，从而使它们变得更清晰并且具有完全显式的常数。还考虑了可以在连续解处评估离散双线性形式的情况，这使得一致性项与到子空间的距离分开出现，从而允许前者独立于后者进行处理。