This paper provides a new abstract stability result for perturbed saddle-point problems which is based on a proper norm fitting. We derive the stability condition according to Babu\v{s}ka's theory from a small inf-sup condition, similar to the famous Ladyzhenskaya-Babu\v{s}ka-Brezzi (LBB) condition, and the other standard assumptions in Brezzi's theory under the resulting combined norm. The proposed framework allows to split the norms into proper seminorms and not only results in simpler (shorter) proofs of many stability results but also guides the construction of parameter robust norm-equivalent preconditioners. These benefits are demonstrated with several examples arising from different formulations of Biot's model of consolidation.
翻译:本文为动荡的马鞍问题提供了一个新的抽象稳定结果,这是基于适当的规范。 根据Babu\v{s{s}ka的理论,我们从一个小的内侧条件(类似于著名的Ladyzhenskaya-Babu\v{s}ka-Brezzi(LBB)条件)和Brezzi理论中其他标准假设(根据由此产生的综合规范)中的其他标准假设(LBB)来得出稳定状态。 拟议的框架允许将规范分为适当的半调,不仅可以产生许多稳定性结果的更简单(更简短)证据,而且还指导了参数稳健的规范等同前提的构建。 这些好处以生物公司合并模式的不同配方所产生的几个例子来证明。