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Lattice approximation for stochastic reaction diffusion equations with one-sided Lipschitz condition


Authors: Martin Sauer and Wilhelm Stannat
Journal: Math. Comp. 84 (2015), 743-766
MSC (2010): Primary 60H15, 60H35; Secondary 35R60, 65C30, 92C20
Published electronically: August 12, 2014
MathSciNet review: 3290962
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Abstract: We consider strong convergence of the finite differences approximation in space for stochastic reaction diffusion equations in one dimension with multiplicative noise under a one-sided Lipschitz condition only. The equation may be additionally coupled with a noisy control variable with global Lipschitz condition but no diffusion. We derive convergence with an implicit rate depending on the regularity of the exact solution. This can be made explicit if the variational solution has more than its canonical spatial regularity. As an application, spatially extended FitzHugh-Nagumo systems with noise are considered.


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Additional Information

Martin Sauer
Affiliation: Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, D-10623 Berlin, Germany
Email: sauer@math.tu-berlin.de

Wilhelm Stannat
Affiliation: Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, D-10623 Berlin, Germany and Bernstein Center for Computational Neuroscience, Philippstr. 13, D-10115 Berlin, Germany
Email: stannat@math.tu-berlin.de

DOI: https://doi.org/10.1090/S0025-5718-2014-02873-1
Keywords: Stochastic reaction diffusion equations, finite difference approximation, FitzHugh-Nagumo system
Received by editor(s): January 27, 2013
Received by editor(s) in revised form: July 22, 2013
Published electronically: August 12, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.