# Splitting up method for the 2D stochastic Navier-Stokes equations

Abstract : In this paper, we deal with the convergence of an iterative scheme for the 2-D stochastic Navier-Stokes Equations on the torus suggested by the Lie-Trotter product formulas for stochastic differential equations of parabolic type. The stochastic system is split into two problems which are simpler for numerical computations. An estimate of the approximation error is given either with periodic boundary conditions. In particular, we prove that the strong speed of the convergence in probability is almost $1/2$. This is shown by means of an $L^2(\Omega,P)$ convergence localized on a set of arbitrary large probability. The assumptions on the diffusion coefficient depend on the fact that some multiple of the Laplace operator is present or not with the multiplicative stochastic term. Note that if one of the splitting steps only contains the stochastic integral, then the diffusion coefficient may not contain any gradient of the solution.
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Article dans une revue
Stochastic Partial Differential Equations : Analysis and Computations, 2014, 2 (4), p.433-470. 〈http://link.springer.com/article/10.1007/s40072-014-0041-7〉
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https://hal.archives-ouvertes.fr/hal-00864579
Contributeur : Annie Millet <>
Soumis le : jeudi 27 février 2014 - 05:43:45
Dernière modification le : jeudi 11 janvier 2018 - 06:12:29
Document(s) archivé(s) le : mardi 27 mai 2014 - 10:51:38

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• HAL Id : hal-00864579, version 2
• ARXIV : 1309.5633

### Citation

Hakima Bessaih, Zdzislaw Brzezniak, Annie Millet. Splitting up method for the 2D stochastic Navier-Stokes equations. Stochastic Partial Differential Equations : Analysis and Computations, 2014, 2 (4), p.433-470. 〈http://link.springer.com/article/10.1007/s40072-014-0041-7〉. 〈hal-00864579v2〉

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