A new framework for the stability analysis of perturbed saddle-point problems and applications in poromechanics

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.