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

Serguei A. Nazarov; Maria Specovius-Neugebauer

Banach Center Publications (2008)

- Volume: 81, Issue: 1, page 297-320
- ISSN: 0137-6934

## Access Full Article

top## Abstract

top## How to cite

topSerguei A. Nazarov, and Maria Specovius-Neugebauer. "Optimal convergence results for the Brezzi-Pitkäranta approximation of the Stokes problem: Exterior domains." Banach Center Publications 81.1 (2008): 297-320. <http://eudml.org/doc/282109>.

@article{SergueiA2008,

abstract = {This paper deals with a strongly elliptic perturbation for the Stokes equation in exterior three-dimensional domains Ω with smooth boundary. The continuity equation is substituted by the equation -ε²Δp + 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 $H^\{5/2-δ\}$ convergence for the velocity part and $H^\{3/2-δ\}$ convergence for the pressure to the solution of the Stokes problem, with δ 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 $H^\{5/2-δ\}_\{loc\}$ and $H^\{3/2-δ\}_\{loc\}$, respectively, of the remainder to the corresponding parts of the Stokes solution.},

author = {Serguei A. Nazarov, Maria Specovius-Neugebauer},

journal = {Banach Center Publications},

keywords = {pressure stabilization method; singular perturbation; asymptotic behavior},

language = {eng},

number = {1},

pages = {297-320},

title = {Optimal convergence results for the Brezzi-Pitkäranta approximation of the Stokes problem: Exterior domains},

url = {http://eudml.org/doc/282109},

volume = {81},

year = {2008},

}

TY - JOUR

AU - Serguei A. Nazarov

AU - Maria Specovius-Neugebauer

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

JO - Banach Center Publications

PY - 2008

VL - 81

IS - 1

SP - 297

EP - 320

AB - This paper deals with a strongly elliptic perturbation for the Stokes equation in exterior three-dimensional domains Ω with smooth boundary. The continuity equation is substituted by the equation -ε²Δp + 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 $H^{5/2-δ}$ convergence for the velocity part and $H^{3/2-δ}$ convergence for the pressure to the solution of the Stokes problem, with δ 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 $H^{5/2-δ}_{loc}$ and $H^{3/2-δ}_{loc}$, respectively, of the remainder to the corresponding parts of the Stokes solution.

LA - eng

KW - pressure stabilization method; singular perturbation; asymptotic behavior

UR - http://eudml.org/doc/282109

ER -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.