Approximation of exterior boundary value problems for the Stokes system

Abstract

Let Ω⊂R³ be an exterior domain and u a solution of the Dirichlet problem for the Stokes system. Let {G_R} 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 u^R of boundary problems which are defined on the bounded subdomain Ω∩G_R. On the external boundary ∂G_R an artificial boundary condition Bu^R=h has to be added. In the three cases – u^R = 0, Tu^R·ν=0 and Tu^R·ν+A·u=0 on ∂G_R – formal asymptotic estimates are derived for u - u^R. 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.