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The inf‐sup constant for the divergence on corner domains
Author(s) -
Costabel Martin,
Crouzeix Michel,
Dauge Monique,
Lafranche Yvon
Publication year - 2015
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.21916
Subject(s) - mathematics , spectrum (functional analysis) , constant (computer programming) , mathematical analysis , constant function , essential spectrum , interval (graph theory) , differential operator , holomorphic function , eigenvalues and eigenvectors , divergence (linguistics) , upper and lower bounds , polygon (computer graphics) , geometry , pure mathematics , combinatorics , physics , piecewise , quantum mechanics , telecommunications , linguistics , philosophy , frame (networking) , computer science , programming language
The inf‐sup constant for the divergence, or LBB constant, is related to the Cosserat spectrum. It has been known for a long time that on nonsmooth domains the Cosserat operator has a nontrivial essential spectrum, which can be used to bound the LBB constant from above. We prove that the essential spectrum on a plane polygon consists of an interval related to the corner angles and that on three‐dimensional domains with edges, the essential spectrum contains such an interval. We obtain some numerical evidence for the extent of the essential spectrum on domains with axisymmetric conical points by computing the roots of explicitly given holomorphic functions related to the corner Mellin symbol. Using finite element discretizations of the Stokes problem, we present numerical results pertaining to the question of the existence of eigenvalues below the essential spectrum on rectangles and cuboids. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 439–458, 2015