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

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.