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 ∞ , p ∞ to the problems in the unbounded domainare the approximating solutions on the truncated domain Ω R , the parameter R controls the exhausting of Ω. The artificial boundary conditions involve the Steklov-Poincaré 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.

The Stokes problem in a periodic layer

“…On the one hand this knowledge is crucial for numerics in the choice of optimal artificial boundary conditions (cf. [14], [15]). On the other hand it is a contribution to the large and highly interesting field of homogenisation techniques relevant for so-called multi-structures, i.e., junctions of thin domains with different limiting dimensions.…”

Steady-state Navier–Stokes flow in an obstructed pipe under mixed boundary conditions and with a prescribed transversal flux rate

“…This is also motivated by the fact that a large variety of problems in Mathematical Fluid Mechanics are studied in unbounded pipes or in a system of unbounded pipes (that is, domains having non-compact boundaries); see in particular the celebrated Leray problem [64], the works of Ladyzhenskaya & Solonnikov [48,51], Amick [4,5], Pileckas et al [37,69] and the recent articles [76,77]. Nevertheless, not only the numerical approximation of these problems must be set in bounded domains (finite pipes or conjunction of finite pipes) [35], also theoretical approaches have been devised in regions with compact boundaries (for example, Leray's argument on the invading domains [45,55]) in order to tackle the original problem, thereby introducing artificial boundary conditions on truncating surfaces, see the articles by Blazy, Nazarov & Specovius-Neugebauer [9,59] and references therein. A second natural question then arises:…”

Modeling of the unsteady flow through a channel with an artificial outflow condition by the Navier–Stokes variational inequality

“…For instance, appropriate boundary conditions on an artificial boundary (for sufficiently large ) are suggested in the paper , where the Oseen system in an exterior domain is studied. In , the authors study the Navier–Stokes equations in a “layer‐like” unbounded domain Ω and they construct approximations of solutions in bounded sub‐domains that satisfy certain artificial conditions (involving local and non‐local boundary operators) on the truncated part of . A similar problem, i.e.…”