A Coupling of BEM and FEM for the Viscous Flow Problem
In this paper we present a combined boundary element method (BEM) and finite element method (FEM) scheme for treating the two-dimensional stationary viscous, incompressible fluid flow past an obstacle. The problem here can be formulated in terms of the dimensionless Navier-Stokes equations with a small parameter, the Reynolds number. This is a typical exterior singular perturbation problem in the sense that there is no solution to the reduced equations (that is, the equations when the parameter is zero) satisfying both the boundary condition and the condition at infinity. As is well known, because of the presence of small parameters, standard numerical schemes, without suitable modifications, are not, in general, applicable for single perturbation problems.
The essential idea of our approach is to divide the region under consideration into two regions, an inner and an outer region. By making use of the asymptotic behavior of the solution to the original problem, we solve actually a transmission problem for different linearized partial differential equations in the inner and outer regions. In the inner region, Stokes equations (see (3)) will be employed, whereas in the outer region, the Oseen equations (see (4)) will be used. The transmission boundary conditions on an auxiliary boundary of the interface resemble the “matching conditions” in singular perturbation theory . By employing Green’s representation for the solution of the Oseen equations in the outer region, a nonlocal boundary condition  will be derived on the auxiliary boundary in terms of appropriate boundary integral operators. With this nonlocal boundary condition on the interface, we then apply the Galerkin method to the Stokes equations in the inner region. In particular, from the asymptotic expansions of the fundamental solution to the Oseen equations, an effective constructive procedure may be developed independent of the small parameter. In principle, our approach follows the works of  and , but we believe it is more appropriate for singular perturbation problems.
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