Nonnegativity preserving convergent schemes for stochastic porous-medium equations
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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.References
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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
- 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
- © Copyright 2018 American Mathematical Society
- 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
- MathSciNet review: 3904138