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

100%
Nazarov S., Specovius-Neugebauer M.
Banach Center Publications
|
2008
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tom 81
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nr 1
297-320
EN
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.
2
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Selfadjoint Extensions for the Elasticity System in Shape Optimization

100%
Nazarov S., Sokołowski J.
Bulletin of the Polish Academy of Sciences. Mathematics
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2004
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tom 52
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nr 3
237-248
EN
Two approaches are proposed to modelling of topological variations in elastic solids. The first approach is based on the theory of selfadjoint extensions of differential operators. In the second approach function spaces with separated asymptotics and point asymptotic conditions are introduced, and a variational formulation is established. For both approaches, accuracy estimates are derived.
3
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Topological derivatives for semilinear elliptic equations

71%
Iguernane M., Nazarov S., Roche J.-R., Sokolowski J., Szulc K.
International Journal of Applied Mathematics and Computer Science
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2009
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tom 19
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nr 2
191-205
EN
The form of topological derivatives for an integral shape functional is derived for a class of semilinear elliptic equations. The convergence of finite element approximation for the topological derivatives is shown and the error estimates in the L∞ norm are obtained. The results of numerical experiments which confirm the theoretical convergence rate are presented.
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