We study operators that project a vector-valued function υ ∈ W1,2(Ω, ℝd) to subspaces formed by the condition that the divergence is orthogonal to a certain amount (finite or infinite) of test functions. The condition that the divergence is equal to zero almost everywhere presents the first (narrowest) limit case while the integral condition of zero mean divergence generates the other (widest) case. Estimates of the distance between υ and the respective projection on such a subspace are important for analysis of various mathematical models related to incompressible media problems (especially in the context of a posteriori error estimates. We establish different forms of such estimates, which contain only local constants associated with the stability (LBB) inequalities for subdomains. The approach developed in the paper also yields two-sided bounds of the inf-sup (LBB) constant.
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References
I. Babuška and A. K. Aziz, Surway Lectures on the Mathematical Foundations of the Finite Element Method, Academic Press, New York (1972).
M. E. Bogovskii, “Solution of the first boundary value problem for the equation of continuity of an incompressible medium,” Soviet Math. Dokl., 248, No. 5, 1037–1040 (1979).
M. Costabel and M. Dauge. “On the inequalities of Babuska–Aziz, Friedrichs and Horgan–Payne,” Arch. Ration. Mech. Anal., 217, No. 3, 873–898 (2015).
M. Costabel, M. Crouzeix, M. Dauge, and Y. Lafranche. “The inf-sup constant for the divergence on corner domains,” Num. Methods PDES, 31, No 2, 439–458 (2015).
M. Dobrowolski, “On the LBB constant on stretched domains,” Math. Nachr., 254/255, 64–67 (2003).
C. Horgan and L. Payne, “On inequalities of Korn, Friedrichs and Babuska–Aziz,” Arch. Ration. Mech. Anal., 82, 165–179 (1983).
M. Kessler, Die Ladyzhenskaya-Konstante in der numerischen Behandlung von Strömungsproblemen, Bayerischen Julius-Maximilians-Universität, Würzburg (2000).
O. A. Ladyzhenskaya, Mathematical Problems in the Dynamics of a Viscous Incompressible Flow, 2nd ed., Gordon and Breach, New York (1969).
O. A. Ladyzenskaja and 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, 81–116 (1976).
J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques, Masson et Cie, Éditeurs, Paris (1967).
M. A. Olshanskii and E. V. Chizhonkov, “On the best constant in the inf sup condition for prolonged rectangular domains,” Mat. Zam., 67, No. 3, 387–396 (2000).
L. E. Payne, “A bound for the optimal constant in an inequality of Ladyzhenskaya and Solonnikov,” IMA J. Appl. Math., 72, 563–569 (2007).
K. I. Piletskas, “On spaces of solenoidal vectors,” Zap. Nauchn. Semin. LOMI, 96, 237–239 (1980).
K. I. Piletskas, “Spaces of solenoidal vectors,” Trudy Mat. Inst. Steklov, 159, 137–149 (1983).
S. Repin, “A posteriori estimates for the Stokes problem,” J. Math. Sci., 109, No. 5, 1950–1964 (2002).
S. Repin, “Estimates of deviations from exact solutions for some boundary–value problems with incompressibility condition,” St. Petersburg Math. J., 16, No. 5, 124–161 (2004).
S. Repin, A Posteriori Estimates for Partial Differential Equations, Walter de Gruyter, Berlin (2008).
S. Repin, “Estimates of deviations from exact solution of the generalized Oseen problem,” Zap. Nauchn. Semin. POMI, 410, 110–130 (2013).
S. Repin, “Estimates of the distance to the set of divergence free fields,” Zap. Nauchn. Semin. POMI, 425, 99–116 (2014).
S. Repin, “On variational representations of the constant in the inf sup condition for the Stokes problem,” Zap. Nauchn. Semin. POMI, 444, 110–123 (2016).
S. Repin, “Estimates of the distance to the set of solenoidal vector fields and applications to a posteriori error control,” Comput. Methods Appl. Math., 15, No. 4, 515–530 (2015).
G. Stoyan, “Towards discrete Velte decompositions and narrow bounds for inf-sup constants,” Comput. Math. Appl., 38, 243–261 (1999).
R. Temam, Navier–Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam (1979).
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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 459, 2017, pp. 83–103.
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Repin, S. On Projectors to Subspaces of Vector-Valued Functions Subject to Conditions of the Divergence-Free Type. J Math Sci 236, 430–445 (2019). https://doi.org/10.1007/s10958-018-4123-3
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DOI: https://doi.org/10.1007/s10958-018-4123-3