Estimates of the distance to the set of divergence free fields

S. Repin

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 44, Tome 425 (2014), pp. 99-116

Résumé

We are concerned with computable estimates of the distance to the set of divergence free fields, which are necessary for quantitative analysis of mathematical models of incompressible media (e.g., Stokes, Oseen, and Navier–Stokes problems). The distance is measured in terms of $L^q$ norm of the gradient with $q\in(1,+\infty)$. For $q=2$, these estimates follow from the so-called inf-sup condition (or Aziz–Babuška–Ladyzhenskaya–Solonnikov inequality) and require sharp estimates of the respective constant, which are known only for a very limited amount of cases. We suggest a way to bypass this difficulty and show that for a vide class of domains (and different boundary conditions) computable estimates of the distance to the set of divergence free field can be presented in the form, which uses inf-sup constants for certain basic problems. In the last section, these estimates are applied to problems in the theory of viscous incompressible fluids. They generate fully computable bounds of the distance to generalized solutions of the problems considered.

Bibliographie

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

[2] I. Babuškam, A. K. Aziz, Survey 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, 359 pp. | MR

[3] 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

[4] M. E. Bogovskii, “Solution of the first boundary value problem for the equation of continuity of an incompressible medium”, Dokl. Akad Nauk SSSR, 248:5 (1979), 1037–1040 | MR

[5] M. E. Bogovskii, “Solution of some vector analysis problems connected with operators Div and Grad”, Trudy Seminar S. L. Sobolev, 1, Akademia Nauk SSSR, Sibirskoe Otdelenie Matematiki, Novosibirsk, 1980, 5–40 (in Russian) ; Dokl. AN SSSR, 248:5 (1979), 1037–1040 | MR | MR

[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] M. Dobrowolski, “On the LBB condition in numerical analysis of the Stokes equations.”, Appl. Numer. Math., 54:3–4 (2005), 314–323 | DOI | MR | Zbl

[9] K. O. Friedrichs, “On certain inequalities and characteristic value problems for analytic functions and for functions of two variables”, Trans. Amer. Math. Soc., 41 (1937), 321–364 | DOI | MR | Zbl

[10] G. P. Galdi, An introduction to the mathematical theory of the Navier–Stokes equations, v. II, Springer Tracts in Natural Philosophy, Linearized steady problems, 39, Springer-Verlag, New York, 1994 | MR | Zbl

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

[12] 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. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 59, 1976, 81–116 (in Russian) | MR | Zbl

[13] D. S. Malkus, “Eigenproblems associated with the discrete LBB-condition for incompressible finite elements”, Int. J. Eng. Sci., 19 (1981), 1299–1310 | DOI | MR | Zbl

[14] J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques, Masson et Cie, Éditeurs, Paris; Academia, Éditeurs, Prague, 1967 | MR | Zbl

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

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

[17] K. I. Piletskas, “Spaces of solenoidal vectors”, Trudy Mat. Inst. Steklov, 159, 1983, 137–149 | MR | Zbl

[18] K. I. Piletskas, “On spaces of solenoidal vectors”, Zap. Nauchn. Semin. LOMI, 96, 1980, 237–239 | MR | Zbl

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

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

[21] S. Repin, “On a posteriori error estimates for the stationary Navier–Stokes problem”, J. Math. Sci. (New York), 150:1 (2008), 1885–1889 | DOI | MR | Zbl

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

[23] S. Repin, “Estimates of deviations from the exact solution of a generalized Oseen problem”, J. Math. Sci. (New York), 195:1 (2013), 64–75 | DOI | MR | Zbl

[24] S. Repin, R. Stenberg, Two-sided a posteriori estimates for the generalized Stokes problem, Preprint A516, Helsinki University of Technology, Institute of Mathematics, 2006

[25] 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

[26] G. Stoyan, “Towards Discrete Velte Decompositions and Narrow Bounds for Inf-Sup Constants”, Computers and Mathematics with Applications, 38 (1999), 243–261 | DOI | MR | Zbl