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Nonnegativity preserving convergent schemes for stochastic porous-medium equations

Authors: Hubertus Grillmeier and Günther Grün
Journal: Math. Comp.
MSC (2010): Primary 35B09, 35K65, 35R35, 37L55, 37M05, 60H15, 65C30, 65N30, 76S05
Published electronically: August 8, 2018
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Abstract: We propose a fully discrete finite-element scheme for stochastic porous-medium equations with linear, multiplicative noise given by a source term. A subtle discretization of the degenerate diffusion coefficient combined with a noise approximation by bounded stochastic increments permits us to prove $ H^1$-regularity and nonnegativity of discrete solutions. By Nikol'skiĭ estimates in time, Skorokhod-type arguments and the martingale representation theorem, convergence of appropriate subsequences towards a weak solution is established. Finally, some preliminary numerical results are presented which indicate that linear, multiplicative noise in the sense of Ito, which enters the equation as a source-term, has a decelerating effect on the average propagation speed of the boundary of the support of solutions.

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  • [1] D. G. Aronson, L. A. Caffarelli, S. Kamin, , and How an initially stationary interface begins to move in porous medium flow, SIAM J. Math. Anal. 14 (1983), no. 4, 639-658. MR 704481,
  • [2] Ľubomír Baňas, Zdzislaw Brzeźniak, Mikhail Neklyudov, and Andreas Prohl, A convergent finite-element-based discretization of the stochastic Landau-Lifshitz-Gilbert equation, IMA J. Numer. Anal. 34 (2014), no. 2, 502-549. MR 3194798,
  • [3] Viorel Barbu and Michael Röckner, Localization of solutions to stochastic porous media equations: finite speed of propagation, Electron. J. Probab. 17 (2012), no. 10, 11. MR 2878789,
  • [4] Viorel Barbu, Giuseppe Da Prato, and Michael Röckner, Existence and uniqueness of nonnegative solutions to the stochastic porous media equation, Indiana Univ. Math. J. 57 (2008), no. 1, 187-211. MR 2400255,
  • [5] Viorel Barbu, Giuseppe Da Prato, and Michael Röckner, Stochastic porous media equations and self-organized criticality, Comm. Math. Phys. 285 (2009), no. 3, 901-923. MR 2470909,
  • [6] Viorel Barbu, Giuseppe Da Prato, and Michael Röckner, Existence of strong solutions for stochastic porous media equation under general monotonicity conditions, Ann. Probab. 37 (2009), no. 2, 428-452. MR 2510012,
  • [7] J. Becker and G. Grün,
    The thin-film equation: Recent advances and some new perspectives,
    J. Phys.: Condensed Matter 17 (2005), 291-307.
  • [8] A. Bensoussan and R. Temam, Équations stochastiques du type Navier-Stokes, J. Functional Analysis 13 (1973), 195-222. MR 0348841
  • [9] Zdzislaw Brzeźniak, Erich Carelli, and Andreas Prohl, Finite-element-based discretizations of the incompressible Navier-Stokes equations with multiplicative random forcing, IMA J. Numer. Anal. 33 (2013), no. 3, 771-824. MR 3081484,
  • [10] Giuseppe Da Prato and Jerzy Zabczyk, Stochastic Equations in Infinite Dimensions, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 152, Cambridge University Press, Cambridge, 2014. MR 3236753
  • [11] B. Davidovitch, E. Moro, and H.A. Stone,
    Spreading of viscous fluid drops on a solid substrate assisted by thermal fluctuations,
    Phys.Rev.Lett. 95 (2005), 244-505.
  • [12] Claude Dellacherie and Paul-André Meyer, Probabilités et potentiel. Chapitres V à VIII, Revised edition, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], vol. 1385, Hermann, Paris, 1980 (French). Théorie des martingales. [Martingale theory]. MR 566768
  • [13] Alexandre Ern and Jean-Luc Guermond, Theory and Practice of Finite Elements, Applied Mathematical Sciences, vol. 159, Springer-Verlag, New York, 2004. MR 2050138
  • [14] Julian Fischer and Günther Grün, Existence of positive solutions to stochastic thin-film equations, SIAM J. Math. Anal. 50 (2018), no. 1, 411-455. MR 3755665,
  • [15] Julian Fischer and Günther Grün, Finite speed of propagation and waiting times for the stochastic porous medium equation: a unifying approach, SIAM J. Math. Anal. 47 (2015), no. 1, 825-854. MR 3313825,
  • [16] Franco Flandoli and Dariusz Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields 102 (1995), no. 3, 367-391. MR 1339739,
  • [17] Benjamin Gess, Strong solutions for stochastic partial differential equations of gradient type, J. Funct. Anal. 263 (2012), no. 8, 2355-2383. MR 2964686,
  • [18] Benjamin Gess, Finite speed of propagation for stochastic porous media equations, SIAM J. Math. Anal. 45 (2013), no. 5, 2734-2766. MR 3101090,
  • [19] Benjamin Gess, Finite time extinction for stochastic sign fast diffusion and self-organized criticality, Comm. Math. Phys. 335 (2015), no. 1, 309-344. MR 3314506,
  • [20] Lorenzo Giacomelli and Günther Grün, Lower bounds on waiting times for degenerate parabolic equations and systems, Interfaces Free Bound. 8 (2006), no. 1, 111-129. MR 2231254,
  • [21] Günther Grün, On the convergence of entropy consistent schemes for lubrication type equations in multiple space dimensions, Math. Comp. 72 (2003), no. 243, 1251-1279. MR 1972735,
  • [22] Günther Grün and Martin Rumpf, Nonnegativity preserving convergent schemes for the thin film equation, Numer. Math. 87 (2000), no. 1, 113-152. MR 1800156,
  • [23] István Gyöngy and Nicolai Krylov, Existence of strong solutions for Itô's stochastic equations via approximations, Probab. Theory Related Fields 105 (1996), no. 2, 143-158. MR 1392450,
  • [24] Nobuyuki Ikeda and Shinzo Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1981. MR 637061
  • [25] Ioannis Karatzas and Steven E. Shreve, Brownian Motion and Stochastic Calculus, 2nd ed., Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1991. MR 1121940
  • [26] Jong Uhn Kim, On the stochastic porous medium equation, J. Differential Equations 220 (2006), no. 1, 163-194. MR 2182084,
  • [27] P. E. Kloeden and E. Platen.
    Numerical Solution of Stochastic Differential Equations.
    Springer, Berlin Heidelberg, 1992.
  • [28] Peter Knabner and Lutz Angermann, Numerical Methods for Elliptic and Parabolic Partial Differential Equations, Texts in Applied Mathematics, vol. 44, Springer-Verlag, New York, 2003. MR 1988268
  • [29] Jacques Simon, Sobolev, Besov and Nikolskiĭ fractional spaces: imbeddings and comparisons for vector valued spaces on an interval, Ann. Mat. Pura Appl. (4) 157 (1990), 117-148. MR 1108473,
  • [30] Aad W. van der Vaart and Jon A. Wellner, Weak Convergence and Empirical Processes, Springer Series in Statistics, Springer-Verlag, New York, 1996. With applications to statistics. MR 1385671
  • [31] Juan Luis Vázquez, The Porous Medium Equation, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. Mathematical theory. MR 2286292
  • [32] Liqing Yan, The Euler scheme with irregular coefficients, Ann. Probab. 30 (2002), no. 3, 1172-1194. MR 1920104,
  • [33] L. Zhornitskaya and A. L. Bertozzi, Positivity-preserving numerical schemes for lubrication-type equations, SIAM J. Numer. Anal. 37 (2000), no. 2, 523-555. MR 1740768,

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Additional Information

Hubertus Grillmeier
Affiliation: Department of Mathematics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstrasse 11, 91058 Erlangen, Germany

Günther Grün
Affiliation: Department of Mathematics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstrasse 11, 91058 Erlangen, Germany

Keywords: Stochastic porous-medium equation, nonnegativity-preserving scheme, stochastic free-boundary problem, martingale solution
Received by editor(s): March 17, 2017
Received by editor(s) in revised form: December 7, 2017, and April 3, 2018
Published electronically: August 8, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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