Approximation of exterior boundary value problems for the Stokes system

Article type: Research Article

Authors: Nazarov, S.A.^; | Specovius-Neugebauer, M.

Affiliations: State Maritime Academy, Kosaya Liniya, 15-A St.-Petersburg, 199026, Russia | Universität Paderborn, 33095 Paderborn, Germany. E-mail: mariasp@uni-paderborn.de

Note: [] This research was supported by the DFG research group “Equations of Hydrodynamics”, Universities of Bayreuth and Paderborn.

Abstract: Let Ω⊂R3 be an exterior domain and u a solution of the Dirichlet problem for the Stokes system. Let {GR} be a set of bounded domains which contain ∂Ω for any R≥1 and exhaust Ω as R→∞. The problem is investigated how u can be approximated by solutions uR of boundary problems which are defined on the bounded subdomain Ω∩GR. On the external boundary ∂GR an artificial boundary condition BuR=h has to be added. In the three cases – uR = 0, TuR·ν=0 and TuR·ν+A·u=0 on ∂GR – formal asymptotic estimates are derived for u - uR. It turns out that the mixed boundary condition leads to the best asymptotic decay if the matrix A(x) is chosen in a proper way. For this boundary condition the unique solvability and R-independent estimates are proved in weighted Sobolev spaces. With these results the formal error estimates are justified rigorously.

DOI: 10.3233/ASY-1997-14302

Journal: Asymptotic Analysis, vol. 14, no. 3, pp. 223-255, 1997