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2004 Optimal results for the Brezzi-Pitkäranta approximation of viscous flow problems
S. A. Nazarov, M. Specovius-Neugebauer
Differential Integral Equations 17(11-12): 1359-1394 (2004).


By introducing the term $-{\varepsilon} ^2 \Delta p$ into the equation of continuity and an additional Neumann boundary condition for the pressure $p$, a strongly elliptic system is obtained which is a singular perturbation of the Stokes system. We use parameter-dependent Sobolev norms to derive asymptotically precise estimates for solutions to the perturbed problem as ${\varepsilon} \searrow 0$. This results in optimal estimates for the difference between solutions to both problems; such estimates are not available by the usually applied energy methods. Under additional regularity assumptions for the data, for the energy estimates, the order of convergence with respect to ${\varepsilon}$ is improved, and convergence in $H^{s+1}$ and $H^s$ norms is obtained for the velocity and pressure with $s\in [0,3/2)$. We verify the asymptotic precision of the estimates by constructing the boundary layers.


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S. A. Nazarov. M. Specovius-Neugebauer. "Optimal results for the Brezzi-Pitkäranta approximation of viscous flow problems." Differential Integral Equations 17 (11-12) 1359 - 1394, 2004.


Published: 2004
First available in Project Euclid: 21 December 2012

zbMATH: 1150.35508
MathSciNet: MR2100032

Primary: 35Q30
Secondary: 35B25 , 76D07 , 76N20

Rights: Copyright © 2004 Khayyam Publishing, Inc.


Vol.17 • No. 11-12 • 2004
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