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

Sauer, Martin; Stannat, Wilhelm

doi:10.1090/S0025-5718-2014-02873-1

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
DOI: https://doi.org/10.1090/S0025-5718-2014-02873-1
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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