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 Babuska's theory from a small inf-sup condition, similar to the famous Ladyzhenskaya-Babuska-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 seminormsand not only results in{\it   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 .
In this paper we prove a new abstract stability result for perturbed saddle-point problems based on a norm fitting technique . We derive the stability condition according to Babuska's theory from a small inf-sup condition, similar to the famous Ladyzhenskaya-Babuska-Brezzi (LBB) condition, and the other standard assumptions in Brezzi's theory , in a combined abstract norm. The construction suggests to form the latter from individual{\it fitted norms that are composed from proper seminorms. This abstract framework not only allows for simpler (shorter) proofs of many stability results but also guides the design of parameter-robust norm-equivalent preconditioners. These benefits are demonstrated on mixed variational formulations of generalized Poisson, Stokes, vector Laplace and Biot's equations .
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which is
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proper norm fitting
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under the resulting combined
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proposed framework allows to split the norms into proper seminormsand not only results in
construction suggests to form the latter from individual
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fitted
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norms that are composed from proper seminorms. This abstract framework not only allows for
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construction of parameter robust
design of parameter-robust
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with several examples arising from different formulations of
on mixed variational formulations of generalized Poisson, Stokes, vector Laplace and
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model of consolidation
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