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Nonnegativity preserving convergent schemes for stochastic porous-medium equations


Authors: Hubertus Grillmeier and Günther Grün
Journal: Math. Comp. 88 (2019), 1021-1059
MSC (2010): Primary 35B09, 35K65, 35R35, 37L55, 37M05, 60H15, 65C30, 65N30, 76S05
DOI: https://doi.org/10.1090/mcom/3372
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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Additional Information

Hubertus Grillmeier
Affiliation: Department of Mathematics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstrasse 11, 91058 Erlangen, Germany
Email: hubertus.grillmeier@fau.de

Günther Grün
Affiliation: Department of Mathematics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstrasse 11, 91058 Erlangen, Germany
Email: gruen@math.fau.de

DOI: https://doi.org/10.1090/mcom/3372
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