## Optimal convergence results for the Brezzi-Pitkäranta approximation of the Stokes problem: Exterior domains

### Volume 81 / 2008

#### Abstract

This paper deals with a strongly elliptic perturbation for the Stokes equation in exterior three-dimensional domains $\Omega$ with smooth boundary. The continuity equation is substituted by the equation $-\varepsilon^2\Delta p + \mathop{\rm div} u =0$, and a Neumann boundary condition for the pressure is added. Using parameter dependent Sobolev norms, for bounded domains and for sufficiently smooth data we prove $H^{5/2-\delta}$ convergence for the velocity part and $H^{3/2-\delta}$ convergence for the pressure to the solution of the Stokes problem, with $\delta$ arbitrarily close to~$0$. For an exterior domain the asymptotic behavior at infinity of the solutions to both problems has also to be taken into account. Although the usual Kondratiev theory cannot be applied to the perturbed problem, it is shown that the asymptotics of the solutions to the exterior Stokes problem and the solution to the perturbed problem coincide completely. For sufficiently smooth data an appropriate decay leads to the convergence of all main asymptotic terms as well as convergence in $H^{5/2-\delta}_{loc}$ and $H^{3/2-\delta}_{loc}$, respectively, of the remainder to the corresponding parts of the Stokes solution.