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.