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IMA Journal of Numerical Analysis Advance Access originally published online on October 3, 2008
IMA Journal of Numerical Analysis 2009 29(3):790-805; doi:10.1093/imanum/drn028
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

The LBB condition in fractional Sobolev spaces and applications

J.-L. Guermond{dagger}

Department of Mathematics, Texas A&M University, 3368 TAMU, College Station, TX 77843-3368, USA

{dagger} Email: guermond{at}math.tamu.edu

Received on 15 December 2006. Revised on 22 March 2008.


  Abstract

The present work focuses on the approximation of the stationary Stokes equations by means of finite-element-like Galerkin methods. It is shown that, provided the velocity space and the pressure space are compatible in some sense, a Ladyzhenskaya–Babuska–Brezzi condition holds in the fractional Sobolev spaces Hs({Omega}), s isin [0, 1]. This result is illustrated in two applications.

Key Words: Stokes operator; finite elements; Navier–Stokes equations; pressure estimates; negative norms; fractional Sobolev spaces

On long leave from Laboratoire d'Informatique pour la Mécanique et les Sciences de l'Ingénieur (LIMSI), BP 133, 91403, Orsay, France.


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