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Zeitschrift für Analysis und ihre Anwendungen


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Volume 27, Issue 2, 2008, pp. 125–155
DOI: 10.4171/ZAA/1348

Published online: 2008-06-30

Artificial Boundary Conditions for the Stokes and Navier–Stokes Equations in Domains that are Layer-Like at Infinity

Sergei A. Nazarov[1] and Maria Specovius-Neugebauer[2]

(1) Institute for Problems in Mechanical Engineering RAS, St. Petersburg, Russian Federation
(2) Universität Kassel, Germany

Artificial boundary conditions are presented to approximate solutions to Stokes- and Navier-Stokes problems in domains that are layer-like at infinity. Based on results about existence and asymptotics of the solutions $v^\infty$, $p^\infty$ to the problems in the unbounded domain $\Omega$ the error $v^\infty-v^R$, $p^\infty-p^R$ is estimated in $H^1(\Omega_R)$ and $L^2(\Omega_R)$, respectively. Here $v^R$, $p^R$ are the approximating solutions on the truncated domain $\Omega_R$, the parameter $R$ controls the exhausting of $\Omega$. The artificial boundary conditions involve the Steklov-Poincar\'{e} operator on a circle together with its inverse and thus turn out to be a combination of local and nonlocal boundary operators. Depending on the asymptotic decay of the data of the problems, in the linear case the error vanishes of order $O(R^{-N})$, where $N$ can be arbitrarily large.

Keywords: Stokes Problem in layers, Navier–Stokes system, artificial boundary conditions, exact boundary conditions, Steklov–Poincaré operator

Nazarov Sergei, Specovius-Neugebauer Maria: Artificial Boundary Conditions for the Stokes and Navier–Stokes Equations in Domains that are Layer-Like at Infinity. Z. Anal. Anwend. 27 (2008), 125-155. doi: 10.4171/ZAA/1348