Approximation of the Stokes Dirichlet Problem in Domains with Cylindrical Outlets

Abstract

Let $\Omega \subset \RR^3$ be a domain with J cylindrical outlets to infinity and u=(v,p) be a solution of the Dirichlet problem for the Stokes system with prescibed flux Hj through the jth outlet. Let $\{\Omega_R\}$ be the set of bounded domains defined by cutting each cylindrical outlet at the distance R from its origin. The problem investigated is how u can be approximated by solutions uR of boundary problems which are defined on the bounded subdomain $\Omega_R$. On the artificial boundary $\po_R \backslash \po $ a boundary condition BuR=h has to be added. By a method similar to the Schwartz' alternating method, the asymptotic behavior (as R tends to infinity) for u-uR is investigated for different types of boundary conditions on the cut cross sections. The existence of solutions uR that are regular up to the edges is shown while using a boundary operator usually related to free boundary problems. For exponentially decaying data asymptotically precise estimates are derived for the difference u-uR; these results hold true for inhomogeneous boundary conditions on the lateral surface $\po$ and nonvanishing divergence. For div v= 0 and homogeneous boundary conditions on $\po$ the case of L2 -forces also is examined.