Geodesic
Estimates of deviations from exact solution of the generalized Oseen problem
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 43, Tome 410 (2013), pp. 110-130 Cet article a éte moissonné depuis la source Math-Net.Ru

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The paper is concerned with a generalized version of the stationary Oseen problem, which often arises in semidiscrete approximation methods used for quantitative analysis of Navier–Stokes equations. We derive a fully computable functional defined for admissible velocity, stress, and pressure fields and prove that this functional generates upper and lower bounds of the total error evaluated in the corresponding combined norm. Moreover, this functional vanishes if and only if its arguments coincide with the exact velocity, stress, and pressure. Therefore, minimization of it is equivalent to solving the Oseen problem. 
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     title = {Estimates of deviations from exact solution of the generalized {Oseen} problem},
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S. Repin. Estimates of deviations from exact solution of the generalized Oseen problem. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 43, Tome 410 (2013), pp. 110-130. http://geodesic.mathdoc.fr/item/ZNSL_2013_410_a4/

[1] I. Babuška, “The finite element method with Lagrangian multipliers”, Numer. Math., 20 (1973), 179–192 | DOI | MR | Zbl

[2] I. 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 | MR

[3] L. Badea, M. Discacciati, A. Quarteroni, “Numerical analysis of the Navier–Stokes/Darcy coupling”, Numerische Mathematik, 115 (2010), 195–227 | DOI | MR | Zbl

[4] J. Bramble, “A proof of the inf-sup condition for the Stokes equations on Lipschitz domains”, Math. Models Methods Appl. Sci., 13:3 (2003), 361–371 | DOI | MR | Zbl

[5] J. Bonvin, M. Picasso, R. Stenberg, “GLS and EVSS methods for a three-field Stokes problem arising from viscoelastic flows”, Comput. Methods Appl. Mech. Engrg., 190 (2001), 3893–3914 | DOI | MR | Zbl

[6] F. Brezzi, “On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers”, R.A.I.R.O. Annal. Numer., R2 (1974), 129–151 | MR | Zbl

[7] M. Dobrowolski, “On the LBB constant on stretched domains”, Math. Nachr., 254–255 (2003), 64–67 | DOI | MR | Zbl

[8] R. Finn, D. R. Smith, “On the linearized hydrodynamical equations in two dimensions”, Arch. Rational Mech. Anal., 25 (1967), 1–25 | DOI | MR | Zbl

[9] M. Fuchs, S. Repin, “Estimates for the deviation from the exact solutions of variational problems modeling certain classes of generalized Newtonian fluids”, Math. Methods Applied Sciences (M2AS), 29 (2006), 2225–2244 | DOI | MR | Zbl

[10] V. Girault, P. A. Raviart, Finite element approximation of the Navier–Stokes equations, Springer, Berlin, 1986 | MR | Zbl

[11] J. G. Heywood, W. Nagata, W. Xie, “A numerically based existence theorem for the Navier–Stokes equations”, J. Math. Fluid Mech., 1 (1999), 5–23 | DOI | MR | Zbl

[12] O. A. Ladyzhenskaya, Mathematical Problems in the Dynamics of a Viscous Incompressible Fluid, Nauka, M., 1970

[13] 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

[14] A. Mikhailov, S. Repin, “Estimates of deviations from exact solution of the Stokes problem in the velocity-vorticity-pressure formulation”, Zap. Nauchn. Semin. POMI, 397, 2011, 73–88

[15] M. A. Olshanskii, E. V. Chizhonkov, “On the best constant in the $inf\,sup$ condition for prolonged rectangular domains”, Mat. Zametki, 67:3 (2000), 387–396 | DOI | MR | Zbl

[16] F. K. G. Oseen, Neuere Methoden und Ergebnisse in der Hydrodynamik, Akademische Velagsgesellschaft, Leipzig, 1927 | Zbl

[17] L. E. Payne, “A bound for the optimal constant in an inequality of Ladyzhenskaya and Solonnikov”, IMA J. Appl. Math., 72 (2007), 563–569 | DOI | MR | Zbl

[18] R. Rannacher, “Finite element methods for the incompressible Navier-Stokes equations”, Fundamental directions in mathematical fluid mechanics, ed. G. P. Galdi, Birkhauser, Basel, 2000, 191–293 | DOI | MR | Zbl

[19] S. Repin, “A posteriori error estimation for variational problems with uniformly convex functionals”, Math. Comput., 69:230 (2000), 481–500 | DOI | MR | Zbl

[20] 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

[21] S. I. Repin, “Aposteriori estimates for the Stokes problem”, J. Math. Sci., 109:5 (2002), 1950–1964 | DOI | MR

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

[23] V. A. Solonnikov, “Estimates for solutions of a non-stationary linearized system of Navier–Stokes equations”, Trudy Mat. Inst. Steklov, 70, 1964, 213–317 | MR | Zbl