Entropy-dissipative discretization of nonlinear diffusion equations and discrete Beckner inequalities

Chainais-Hillairet, Claire; Jüngel, Ansgar; Schuchnigg, Stefan

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 1, pp. 135-162.

The time decay of fully discrete finite-volume approximations of porous-medium 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.

N.D. Alikakos and R. Rostamian, Large time behavior of solutions of Neumann boundary value problem for the porous medium equation. Indiana Univ. Math. J. 30 (1981) 749–785. | DOI | MR | Zbl

T. Arbogast, M. Wheeler and N.-Y. Zhang, A nonlinear mixed finite element method for a degenerate parabolic equation arising in flow in porous medium. SIAM J. Numer. Anal. 33 (1996) 1669–1687. | DOI | MR | Zbl

A. Arnold, P. Markowich, G. Toscani and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations. Commun. Partial Differ. Equ. 26 (2001) 43–100. | DOI | MR | Zbl

J. Barrett, J. Blowey and H. Garcke, Finite element approximation of a fourth order nonlinear degenerate parabolic equation. Numer. Math. 80 (1998) 525–556. | DOI | MR | Zbl

W. Beckner, A generalized Poincaré inequality for Gaussian measures. Proc. Amer. Math. Soc. 105 (1989) 397–400. | MR | Zbl

J. Berryman, Extinction time for fast diffusion. Phys. Lett. A 72 (1979) 107–110. | DOI

M. Bessemoulin-Chatard, C. Chainais–Hillairet and F. Filbet, On discrete functional inequalities for some finite volume schemes. IMA J. Numer. Anal. 35 (2015) 1125–1149. | DOI | MR | Zbl

M. Bonforte and G. Grillo, Asymptotics of the porous media equations via Sobolev inequalities. J. Funct. Anal. 225 (2005) 33–62. | DOI | MR | Zbl

M. Bonforte, J. Dolbeault, G. Grillo and J.L. Vázquez, Sharp rates of decay of solutions to the nonlinear fast diffusion equation $via$ functional inequalities. Proc. Natl. Acad. Sci. USA 107 (2010) 16459–16464. | DOI | MR | Zbl

J.A. Carrillo and G. Toscani, Asymptotic $L^1$-decay of solutions of the porous medium equation to selfsimilarity. Indiana Univ. Math. J. 49 (2000) 113–141. | DOI | MR | Zbl

J.A. Carrillo, A. Jüngel, P. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities. Monatsh. Math. 133 (2001) 1–82. | DOI | MR | Zbl

J.A. Carrillo, A. Jüngel and S. Tang, Positive entropic schemes for a nonlinear fourth-order equation. Discrete Contin. Dyn. Syst. B 3 (2003) 1–20. | MR | Zbl

J.A. Carrillo, J. Dolbeault, I. Gentil, and A. Jüngel, Entropy-energy inequalities and improved convergence rates for nonlinear parabolic equations. Discr. Cont. Dyn. Syst. B 6 (2006) 1027–1050. | MR | Zbl

F. Cavalli, G. Naldi, G. Puppo and M. Semplice, High-order relaxation schemes for nonlinear degenerate diffusion problems. SIAM J. Numer. Anal. 45 (2007) 2098–2119. | DOI | MR | Zbl

M. Del Pino and J. Dolbeault, Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions. J. Math. Pures Appl. 81 (2002) 847–875. | DOI | MR | Zbl

J. Denzler and R. Mccann, Fast diffusion to self-similarity: complete spectrum, long-time asymptotics, and numerology. Arch. Ration. Mech. Anal. 175 (2005) 301–342. | DOI | MR | Zbl

J. Dolbeault, I. Gentil, A. Guillin and F.-Y. Wang, $L^q$-functional inequalities and weighted porous media equations. Potential Anal. 28 (2008) 35–59. | DOI | MR | Zbl

J. Dolbeault, B. Nazaret and G. Savaré, On the Bakry-Emery criterion for linear diffusions and weighted porous media equations. Commun. Math. Sci. 6 (2008) 477–494. | DOI | MR | Zbl

C. Ebmayer and W.W. Liu, Finite element approximation of the fast diffusion and the porous medium equations. SIAM J. Numer. Anal. 46 (2008) 2393–2410. | DOI | MR | Zbl

F. Edgeworth, On the probable errors of frequency-constants (contd.). J. Roy. Stat. Soc. 71 (1908) 499–512. | DOI

R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods. In Vol. 7 of Handbook of Numerical Analysis, edited by P.G. Ciarlet and J.L. Lions. North-Holland, Amsterdam (2000) 713–1020. | MR | Zbl

R. Eymard, T. Gallouët, R. Herbin and A. Michel, Convergence of a finite volume scheme for nonlinear degenerate parabolic equations. Numer. Math. 92 (2002) 41–82. | DOI | MR | Zbl

R. Eymard, M. Gutnic and D. Hilhorst, The finite Volume Method for an elliptic-parabolic equation. Proc. of the Algoritmy’97 Conference on Scientific Computing (Zuberec). Acta Math. Univ. Comenian. (N.S.) 67 (1998) 181–195. | MR | Zbl

A. Glitzky and K. Gärtner, Energy estimates for continuous and discretized electro-reaction-diffusion systems. Nonlin. Anal. 70 (2009) 788–805. | DOI | MR | Zbl

G. Grillo and M. Muratori, Sharp short and long-time asymptotics for solutions to the Neumann problem for the porous media equation. J. Differ. Equ. 254 (2013) 2261–2288. | DOI | MR | Zbl

G. Grillo and M. Muratori, Sharp asymptotics for the porous media equation in low dimensions via Gagliardo–Nirenberg inequalities. Riv. Mat. Univ. Parma 5 (2014) 15–38. | MR | Zbl

L. Gross, Logarithmic Sobolev inequalities. Amer. J. Math. 97 (1975) 1061–1083. | DOI | MR | Zbl

G. Grün and M. Rumpf, Nonnegativity preserving convergent schemes for the thin film equation. Numer. Math. 87 (2000) 113–152. | DOI | MR | Zbl

A. Jüngel and D. Matthes, An algorithmic construction of entropies in higher-order nonlinear PDEs. Nonlinearity 19 (2006) 633–659. | DOI | MR | Zbl

A. Jüngel and D. Matthes, The Derrida–Lebowitz–Speer–Spohn equation: existence, non-uniqueness, and decay rates of the solutions. SIAM J. Math. Anal. 39 (2008) 1996–2015. | DOI | MR | Zbl

A. Jüngel and R. Pinnau, A positivity preserving numerical scheme for a nonlinear fourth-order parabolic equation. SIAM J. Numer. Anal. 39 (2001) 385–406. | DOI | MR | Zbl

J. King, Interacting dopant diffusions in crystalline silicon. SIAM J. Appl. Math. 48 (1988) 405–415. | DOI | MR | Zbl

R. Latala and K. Oleszkiewicz, Between Sobolev and Poincaré. Lect. Notes Math. 1745 (2000) 147–168. | DOI | MR | Zbl

M. Ledoux, On Talagrand’s deviation inequalities for product measures. ESAIM: PS 1 (1996) 63–87. | DOI | Numdam | MR | Zbl

E. Lieb and M. Loss, Analysis, 2nd edition. Amer. Math. Soc. Providence (2010). | MR | Zbl

P.-L. Lions and S. Mas-Gallic, Une méthode particulaire déterministe pour des équations diffusives non linéaires. C. R. Acad. Sci. Paris Sér. I Math 332 (2001) 369–376. | DOI | MR | Zbl

A. Michel and J. Vovelle, Entropy formulation for parabolic degenerate equations with general Dirichlet boundary conditions and application to the convergence of FV methods. SIAM J. Numer. Anal. 41 (2003) 2262–2293. | DOI | MR | Zbl

M. Ohlberger, A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations. Math. Mod. Numer. Anal. 35 (2001) 355–387. | DOI | Numdam | MR | Zbl

M. Rose, Numerical methods for flows through porous media I. Math. Comput. 40 (1983) 435–467. | DOI | MR | Zbl

C. Seis, Long-time asymptotics for the porous medium equation: the spectrum of the linearized operator. J. Differ. Equ. 256 (2014) 1191–1223. | DOI | MR | Zbl

O. Shisha, On the discrete version of Wirtinger’s inequality. Amer. Math. Monthly 80 (1973) 755–760. | DOI | MR | Zbl

J.L. Vázquez, Smoothing and Decay Estimates for Nonlinear Parabolic Equations of Porous Medium Type. Vol. 33 of Oxford Lect. Ser. Math. Appl. Oxford University Press (2006). | MR | Zbl

J.L. Vázquez, The Porous Medium Equation. Mathematical Theory. Oxford University Press, Oxford (2007). | MR | Zbl

Z. Wu, J. Yin and C. Wang, Elliptic and Parabolic Equations. World Scientific, Singapore (2006). | Zbl

L. Zhornitskaya and A. Bertozzi, Positivity-preserving numerical schemes for lubrication-type equations. SIAM J. Numer. Anal. 37 (2000) 523–555. | DOI | MR | Zbl