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C. Chainais-Hillairet, A. Jüngel, S. Schuchnigg:
"Entropy-dissipative discretization of nonlinear diffusion equations and discrete Beckner inequalities";
in: "ASC Report 11/2013", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2013, ISBN: 978-3-902627-06-3, 1 - 31.

The time decay of fully discrete finite-volume approximations of porousmedium and fast-diffusion equations with Neumann or periodic boundary conditions is proved in the entropy sense. The algebraic or exponential decay rates are computed explicitly.
In particular, the numerical scheme dissipates all zeroth-order entropies which are dissipated by the continuous equation. The proofs are based on novel continuous and discrete generalized Beckner inequalities. Furthermore, the exponential decay of some
first-order entropies is proved in the continuous and discrete case using systematic integration by parts. Numerical experiments in one and two space dimensions illustrate the theoretical results and indicate that some restrictions on the parameters seem to be only
technical.

Porous-medium equation, fast-diffusion equation, finite-volume method, entropy dissipation, Beckner inequality, entropy construction method.

http://www.asc.tuwien.ac.at/preprint/2013/asc11x2013.pdf