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On discrete functional inequalities for some finite volume schemes. (English) Zbl 1326.65145
For finite volume schemes on an open, bounded, polyhedral subset \(\Omega\) of \(\mathbb R^N\) (for \(N\geq 2\)), the authors prove discrete counterparts (i.e., for functions defined on a (regular) mesh in \(\Omega\) which are constant on every single element of the mesh) of the Poincaré-Sobolev and the Gagliardo-Nirenberg-Sobolev inequalities which are useful for the convergence analysis of discrete approximate solutions to boundary value problems for partial differential equations. For their investigation, the continuous embedding from the functional space \(BV(\Omega)\) into \(L^{N/(N-1)}(\Omega)\) is important, see, e.g. [W. P. Ziemer, Weakly differentiable functions. Sobolev spaces and functions of bounded variation. Graduate Texts in Mathematics, 120. Berlin etc.: Springer-Verlag (1989; Zbl 0692.46022)], and a corresponding trace relation by L. C. Evans and R. F. Gariepy [Measure theory and fine properties of functions. 2nd revised ed. Textbooks in Mathematics. Boca Raton, FL: CRC Press (2015; Zbl 1310.28001)]. A special construction is needed for the case of homogeneous Dirichlet conditions.
Finally, the results are specialized to discrete duality finite volume approximations in which, along with the basic and the dual mesh for the discrete solution, a diamond mesh is exploited to approximate the gradients.

MSC:
65N08 Finite volume methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
26D10 Inequalities involving derivatives and differential and integral operators
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