Geodesic
On the Stokes problem with model boundary conditions
Sbornik. Mathematics, Tome 188 (1997) no. 4, pp. 603-620 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

For a wide class of two- and three-dimensional domains, boundary conditions on the velocity vector field for the Stokes problem are indicated ensuring that the corresponding Schur complement is the identity operator. These boundary conditions make it possible to “decouple” the Stokes problem into two separate problems, for pressure and for velocity. The solubility of the problem and the regularity of its solutions are studied and the connections between the results obtained and certain aspects of numerical methods in hydrodynamics (such as the LBB condition and the numerical solution of the generalized Stokes problem) are considered. 
@article{SM_1997_188_4_a3,
     author = {M. A. Ol'shanskii},
     title = {On the {Stokes} problem with model boundary conditions},
     journal = {Sbornik. Mathematics},
     pages = {603--620},
     year = {1997},
     volume = {188},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1997_188_4_a3/}
}
TY  - JOUR
AU  - M. A. Ol'shanskii
TI  - On the Stokes problem with model boundary conditions
JO  - Sbornik. Mathematics
PY  - 1997
SP  - 603
EP  - 620
VL  - 188
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/SM_1997_188_4_a3/
LA  - en
ID  - SM_1997_188_4_a3
ER  - 
%0 Journal Article
%A M. A. Ol'shanskii
%T On the Stokes problem with model boundary conditions
%J Sbornik. Mathematics
%D 1997
%P 603-620
%V 188
%N 4
%U http://geodesic.mathdoc.fr/item/SM_1997_188_4_a3/
%G en
%F SM_1997_188_4_a3
M. A. Ol'shanskii. On the Stokes problem with model boundary conditions. Sbornik. Mathematics, Tome 188 (1997) no. 4, pp. 603-620. http://geodesic.mathdoc.fr/item/SM_1997_188_4_a3/

[1] Temam R., Uravneniya Nave–Stoksa. Teoriya i chislennyi analiz, Mir, M., 1981 | MR | Zbl

[2] Gresho P. M., Sani R. L., “On pressure boundary conditions for incompressible Navier–Stokes equations”, Internat. J. Numer. Methods Fluids, 7 (1987), 1111–1145 | DOI | Zbl

[3] Gresho P. M., “Incompressible fluid dynamics: some fundamental formulatio issues”, Annual Review of Fluid Mechanics, 23 (1991), 413–453 | DOI | MR | Zbl

[4] Brezzi F., Fortin M., Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991 | MR | Zbl

[5] Chizhonkov E. V., “Application of the Cossera spectrum to the optimization of a method for solving the Stokes problem”, Russian J. Numer. Anal. Math. Modelling, 9:3 (1994), 191–199 | MR | Zbl

[6] Crouzeix M., “Etude d'une methode de linearisation. Resolution des equations de Stokes stationaries. Application aux equations des Navier–Stokes stationaries”, Cahiere de l'IRIA, 1974, no. 12, 139–244

[7] Mikhlin S. G., “Spektr puchka operatorov teorii uprugosti”, UMN, 28:3 (1973), 43–82 | MR | Zbl

[8] Kobelkov G. M., “O chislennykh metodakh resheniya uravnenii Nave–Stoksa v peremennykh skorost-davlenie”, Vychislitelnye protsessy i sistemy, no. 8, Nauka, M., 1991, 204–237 | MR

[9] Olshanskii M. A., “Ob odnoi zadache tipa Stoksa s parametrom”, ZhVM i MF, 36:2 (1996), 75–86 | MR

[10] Aristov P. P., Chizhonkov E. V., On the constant in the LBB condition for rectangular domains, Report No 9534, Dep. Math. Univ. Nijmegen, 1995 | MR

[11] Bakhvalov N. S., “Solution of the Stokes nonstationary problems by the fictious domain method”, Russian J. Numer. Anal. Math. Modelling, 10:3 (1995), 163–172 | MR | Zbl

[12] Kobel'kov G. M., “On solving the Navier–Stokes equations at large Reynolds numbers”, Russian J. Numer. Anal. Math. Modelling, 10:1 (1995), 33–40 | MR | Zbl

[13] Kobel'kov G. M., To numerical solution of nonstationary Stokes problem, Report No 9536, Dep. Math. Univ. Nijmegen, 1995

[14] Ol'shanskii M. A., “On numerical solution of nonstationary Stokes equations”, Russian J. Numer. Anal. Math. Modelling, 10:1 (1995), 81–92 | MR | Zbl

[15] Girault V., Raviart P. A., Finite element methods for Navier–Stokes equations, Springer-Verlag, Berlin, 1986 | MR | Zbl

[16] Nečas J., Equations aux derivees partielles, Presses de l'Universite de Montreal, Montreal, 1965

[17] Kobel'kov G. M., Valedinskiy V. D., “On the inequality $\|p\|\le c\|\operatorname{grad}p\|_{-1}$ and its finite dimensional image”, Soviet J. Numer. Anal. Math. Modelling, 1:3 (1986), 189–200 | MR | Zbl

[18] Grisvard P., Elliptic Problems in Nonsmooth Domains, Pitman Publ., Boston, 1985 | MR | Zbl

[19] Cattabriga L., “Su un problema al contorno relativo al sistema di equazioni di Stokes”, Sem. Mat. Univ. Padova, 31 (1961), 308–340 | MR | Zbl

[20] Foias C., Temam R., “Remarques sur les equations de Navier–Stokes et les phenomenes successifs de bifurcation”, Ann. Scuola Norm. Sup. Pisa Cl. Sci (4), 5:1 (1978), 29–63 | MR | Zbl

[21] Nikolskii S. M., Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, Nauka, M., 1969 | MR

[22] Gunsburger M. D., Finite element methods for viscous incompressible flows, Acad. Press, INC, 1989

[23] Cahouet J., Chabart J. P., “Some fast 3D finite element solvers for the generalized Stokes problem”, Internat. J. Numer. Methods Fluids, 8 (1988), 869–895 | DOI | MR