The inf-sup constant for the divergence on corner domains

Martin Costabel Michel Crouzeix, Monique Dauge, Yvon Lafranche

February 2014. Prépublication IRMAR, 14-08.

Numerical Methods for Partial Differential Equations, 31, Issue 2, 439–458 (2015).
On line DOI: 10.1002/num.21916 (2014).

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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 non-smooth domains the Cosserat operator has a non-trivial 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.