Approximation of exterior boundary value problems for the Stokes system
С. А. Назаров, Maria Specovius‐Neugebauer · Asymptotic Analysis · 1997
Let Ω⊂R 3 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.