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Zap. Nauchn. Sem. POMI, 2017, Volume 459, Pages 83–103 (Mi znsl6466)  

On projectors to subspaces of vector valued functions subject to conditions of the divergence free type

S. Repinab

a St. Petersburg Department of V. A. Steklov Institute of Mathematics of the Russian Academy of Sciences, 191011, Fontanka 27, St. Petersburg
b Peter the Great St. Petersburg Polytechnic University, St. Petersburg

Abstract: We study operators projecting a vector valued function $v\in W^{1,2}(\Omega,\mathbb Rd)$ to subspaces formed by the condition that the divergence is orthogonal to a certain amount (finite or infinite) of test functions. The condition that 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 $v$ and the respective projection $\mathsf P_\mathbb Sv$ 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, see [15–17]. 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.

Key words and phrases: divergence free functions, mathematical models of incompressible media, the LBB condition.

Funding Agency Grant Number
Russian Foundation for Basic Research 17-01-00099-a
The research was partially supported by RFBR grant No. 17-01-00099-a.


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UDC: 517
Received: 30.09.2017
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Citation: S. Repin, “On projectors to subspaces of vector valued functions subject to conditions of the divergence free type”, Boundary-value problems of mathematical physics and related problems of function theory. Part 46, Zap. Nauchn. Sem. POMI, 459, POMI, St. Petersburg, 2017, 83–103

Citation in format AMSBIB
\Bibitem{Rep17}
\by S.~Repin
\paper On projectors to subspaces of vector valued functions subject to conditions of the divergence free type
\inbook Boundary-value problems of mathematical physics and related problems of function theory. Part~46
\serial Zap. Nauchn. Sem. POMI
\yr 2017
\vol 459
\pages 83--103
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl6466}


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