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

### Sergei A. Nazarov

Institute for Problems in Mechanical Engineering RAS, St. Petersburg, Russian Federation### Maria Specovius-Neugebauer

Universität Kassel, Germany

## Abstract

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.

## Cite this article

Sergei A. Nazarov, Maria Specovius-Neugebauer, Artificial Boundary Conditions for the Stokes and Navier–Stokes Equations in Domains that are Layer-Like at Infinity. Z. Anal. Anwend. 27 (2008), no. 2, pp. 125–155

DOI 10.4171/ZAA/1348