Approximation of the Stokes Dirichlet Problem in Domains with Cylindrical Outlets

Specovius–Neugebauer, Maria

doi:10.1137/s0036141097325083

Cited by 20 publications

(23 citation statements)

References 26 publications

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“…The last two integrals vanish according to (30), while the remainder gets small as R tends to inÿnity-as we will see in the proof of Theorem 4.1.…”

Section: Note Thatmentioning

confidence: 67%

“…Another way to succeed here is to search for u R as a solution to the problem (77) with F = X R f| R , G = X R g and H = 0, here X R is a smooth cut-o function which vanishes near the edge (cf. Reference [30]). With some additional technical e orts, it is possible to prove the existence of a unique solution u R and error estimates similar to (71).…”

Section: Remark 43 (The Case Of Non-zero Boundary Data)mentioning

confidence: 99%

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Artificial boundary conditions for Petrovsky systems of second order in exterior domains and in other domains of conical type

Назаров

Specovius–Neugebauer

2004

Math Methods in App Sciences

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SUMMARYThe approximation of solutions to boundary value problems on unbounded domains by those on bounded domains is one of the main applications for artiÿcial boundary conditions. Based on asymptotic analysis, here a new method is presented to construct local artiÿcial boundary conditions for a very general class of elliptic problems where the main asymptotic term is not known explicitly. Existence and uniqueness of approximating solutions are proved together with asymptotically precise error estimates. One class of important examples includes boundary value problems for anisotropic elasticity and piezoelectricity.

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“…The last two integrals vanish according to (30), while the remainder gets small as R tends to inÿnity-as we will see in the proof of Theorem 4.1.…”

Section: Note Thatmentioning

confidence: 67%

Section: Remark 43 (The Case Of Non-zero Boundary Data)mentioning

confidence: 99%

Artificial boundary conditions for Petrovsky systems of second order in exterior domains and in other domains of conical type

Назаров

Specovius–Neugebauer

2004

Math Methods in App Sciences

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“…[8], [17], [18], [24]- [33]). Most of these papers are restricted to stationary systems, whereas instationary systems have been less studied.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Stokes resolvent systems in an infinite cylinder

Farwig

2007

Mathematische Nachrichten

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In an infinite cylinder Ω = Σ × R, where Σ ⊂ R n−1 , n ≥ 3, is a bounded domain of C 1,1 class, we study the unique solvability of Stokes resolvent systems in L q (R; L 2 (Σ)) for 1 < q < ∞ and in vector-valuedBy a partial Fourier transform along the axis of the cylinder Ω the given system is reduced to a parametrized system on Σ, for which parameter independent estimates are proved. For further applications we obtain even parameter independent estimates in L r (Σ), 1 < r < ∞, in the non-solenoidal case prescribing an arbitrary divergence g = div u. Then operator-valued multiplier theorems are used for the final estimates of the Stokes resolvent systems in Ω.

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“…Their choice is based on the asymptotic behavior of solutions at infinity. In particular, for elliptic boundary value problems in exterior domains and domains with cylindrical or conical outlets to infinity, ABCs in differential form were systematically developed during the last decades (see e.g., [1,2,4,5,7,9,10,14,23,24,32,34]) and the papers quoted there.) The common feature of local ABCs are estimates for the truncation error of the form u ∞ − u R = O(R −γ ) as R tends to infinity, with some γ > 0.…”

Section: Introductionmentioning

confidence: 99%

Artificial Boundary Conditions for the Stokes and Navier–Stokes Equations in Domains that are Layer-Like at Infinity

Назаров¹,

Specovius–Neugebauer²

2008

Z. Anal. Anwend.

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Artificial boundary conditions are presented to approximate solutions to Stokes-and Navier-Stokes problems in domains that are layer-like at infinity. Based on results about existence and asymptotics of the solutions v ∞ , p ∞ to the problems in the unbounded domainare the approximating solutions on the truncated domain Ω R , the parameter R controls the exhausting of Ω. The artificial boundary conditions involve the Steklov-Poincaré operator on a circle together with its inverse and thus turn out to be a combination of local and nonlocal boundary operators. Depending on the asymptotic decay of the data of the problems, in the linear case the error vanishes of order O(R −N ), where N can be arbitrarily large.

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Approximation of the Stokes Dirichlet Problem in Domains with Cylindrical Outlets

Cited by 20 publications

References 26 publications

Artificial boundary conditions for Petrovsky systems of second order in exterior domains and in other domains of conical type

Artificial boundary conditions for Petrovsky systems of second order in exterior domains and in other domains of conical type

Stokes resolvent systems in an infinite cylinder

Artificial Boundary Conditions for the Stokes and Navier–Stokes Equations in Domains that are Layer-Like at Infinity

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