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Register a new userA new framework for the stability analysis of perturbed saddle-point problems and applications
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Added by
Qingguo Hong
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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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 Babuv{s}kas theory from a small inf-sup condition, similar to the famous Ladyzhenskaya-Babuv{s}ka-Brezzi (LBB) condition, and the other standard assumptions in Brezzis 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 Biots equations.
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In this paper, two types of Schur complement based preconditioners are studied for twofold and block tridiagonal saddle point problems. One is based on the nested (or recursive) Schur complement, the other is based on an additive type Schur complement after permuting the original saddle point systems. We discuss different preconditioners incorporating the exact Schur complements. It is shown that some of them will lead to positive stable preconditioned systems. Our theoretical analysis is instructive for devising various exact and inexact preconditioners, as well as iterative solvers for many twofold and block tridiagonal saddle point problems.
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