Geodesic
Estimates of deviations from exact solution of the Stokes problem in the vorticity-velocity-pressure formulation
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 42, Tome 397 (2011), pp. 73-88 
A. Mikhaylov; S. Repin. Estimates of deviations from exact solution of the Stokes problem in the vorticity-velocity-pressure formulation. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 42, Tome 397 (2011), pp. 73-88. http://geodesic.mathdoc.fr/item/ZNSL_2011_397_a3/
@article{ZNSL_2011_397_a3,
     author = {A. Mikhaylov and S. Repin},
     title = {Estimates of deviations from exact solution of the {Stokes} problem in the vorticity-velocity-pressure formulation},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {73--88},
     year = {2011},
     volume = {397},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_397_a3/}
}
TY  - JOUR
AU  - A. Mikhaylov
AU  - S. Repin
TI  - Estimates of deviations from exact solution of the Stokes problem in the vorticity-velocity-pressure formulation
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2011
SP  - 73
EP  - 88
VL  - 397
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2011_397_a3/
LA  - en
ID  - ZNSL_2011_397_a3
ER  - 
%0 Journal Article
%A A. Mikhaylov
%A S. Repin
%T Estimates of deviations from exact solution of the Stokes problem in the vorticity-velocity-pressure formulation
%J Zapiski Nauchnykh Seminarov POMI
%D 2011
%P 73-88
%V 397
%U http://geodesic.mathdoc.fr/item/ZNSL_2011_397_a3/
%G en
%F ZNSL_2011_397_a3

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Vorticity-velocity-pressure formulation for the stationary Stokes problem in 2D is considered. We analyze the corresponding generalized formulation, establish sufficient conditions that guarantee existence of the generalized solution and deduce estimates of the difference between the exact solution (i.e., exact velocity, vorticity, and pressure) and an arbitrary approximating function (velocity, vorticity, pressure) that belongs to the corresponding functional class and satisfies the boundary conditions. For this purpose we use the method suggested in [10, 12], which is based on transformations of the integral identity that defines the corresponding generalized solution.

[1] M. Amara, E. Chacon Vera, D. Trujillo, “Vorticity-velocity-pressure formulation for Stokes problem”, Math Comput., 73:248 (2003), 1673–1697 | DOI | MR

[2] Babuška, A. K. Aziz, “Surway lectures on the mathematical foundations of the finite element method”, The mathematical formulations of the finite element method with applications to partial differential equations, Academic Press, New York, 1972, 5–359 | MR

[3] C. Conca, C. Pares, O. Pironneau, M. Thiriet, “Navier–Stokes equations with imposed pressure and velocity fluxes”, Int. J. Numer. Methods Fluids, 20 (1995), 267–287 | DOI | MR | Zbl

[4] M. Fuchs, S. Repin, “Estimates, of thedeviations from the exact solutions for variational inequalities describing the stationary flow of certain viscous incompressible fluids”, Math. Methods Appl. Sci., 33:9 (2010), 1136–1147 | MR | Zbl

[5] V. A. Kondrat'ev, “Boundary-value problems for elliptic equations in domains with conical or angular points”, Trans. Mosc. Math. Soc., 16 (1967), 209–292 | MR | Zbl

[6] O. A. Ladyzenskaja, V. A. Solonnikov, “Some problems of vector analysis and generalized formulations of boundary-value problems for the Navier–Stokes equation”, Zap. Nauchn. Semin. LOMI, 59, 1976, 81–116 | MR | Zbl

[7] V. Maz'ya, S. Nazarov, B. Plamenevskij, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains, v. 1, Birkhauser, Basel, 2000

[8] O. A. Ladyzhenskaya, Mathematical problems in the dynamics of a viscous incompressible fluid, Nauka, Moscow, 1970

[9] P. Neittaanmaki, S. Repin, “A posteriori error majorants for approximations of the evolutionary Stokes problem”, J. Numer. Math., 18:2 (2010), 119–134 | DOI | MR | Zbl

[10] S. Repin, “Two-sided estimates of deviation from exact solutions of uniformly elliptic equations”, Proc. St. Petersburg Math. Soc., 9, 2001, 143–171 | MR | Zbl

[11] S. Repin, “Estimates of deviations from exact solutions for some boundary-value problems with incompressibility condition”, Algebra Analiz, 16:5 (2004), 124–161 | MR | Zbl

[12] S. Repin, A posteriori estimates for partial differential equations, Walter de Gruyter, Berlin, 2008 | MR

[13] S. Repin, R. Stenberg, “Two-sided a posteriori error estimates for the generalized Stokes problem”, J. Math. Sci. (New York), 142 (2007), 1828–1843 | DOI | MR | Zbl