On the existence uniqueness regularity and approximation of the exterior Stokes problem in R³⁺

Conditions under which a solution exists and is unique for the stationary Stokes problem in exterior domains in R³ are discussed. The domains considered are the complements in R³ of bounded star shaped sets and the class of functions for which existence is displayed have "physically reasonable" behavior at infinity. This class is characterized by a finite Dirichlet integral. We use a method that employs a variational formulation of the problem and avoids the use of a Helmholtz decomposition of vector fields in unbounded domains.

Then we show that a problem on a finite domain denoted by Ω_R approximates the problem on the unbounded domain Ω. An artificial boundary δΩ_R is introduced and different boundary conditions are imposed. We study two cases of artificial boundary conditions, namely the zero velocity boundary condition and the zero stress boundary condition. Approximation results are dicussed. In particular, we emphasize the case where the body force belong to L²(Ω) and has compact support in Ω.

Next, we treat the finite element approximation of the truncated problem. The case of zero velocity artificial boundary condition is treated separately from the case of zero stress boundary condition. Error estimates for the different cases are derived. The final estimates between the problem on the unbounded domain and the discrete problem reduce to a two parameter approximation. To balance the truncation error with the discretization error, we consider a mesh grading technique.

Finally, we treat a truncated problem in which we impose exact boundary condtions at the artificial boundary δΩ_R. This is achieved by studying the physical conditions associated with the hydrodynamic equilibrium of the fluid patches inside and outside Ω_R. Actually, the solution, outside the support of the body forces, can be written explicitly in terms of a boundary integral involving the velocity and the stresses at the boundary. Existence and uniqueness of the coupled problem is studied as well as finite element approximations which turn out to be one parameter approximations.