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Numerical approximation of the stochastic Navier–Stokes equations through artificial compressibility

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Abstract

A constructive numerical approximation of the two-dimensional unsteady stochastic Navier–Stokes equations of an incompressible fluid is proposed via a pseudo-compressibility technique involving a penalty parameter \(\varepsilon \). Space and time are discretized through a finite element approximation and an Euler method. The convergence analysis of the suggested numerical scheme is investigated throughout this paper. It is based on a local monotonicity property permitting the convergence toward the unique strong solution of the stochastic Navier–Stokes equations to occur within the originally introduced probability space.

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Acknowledgements

The author wants to thank Prof. Ludovic Goudenège (Ecole CentraleSupélec) for his valuable comments, Prof. Andreas Prohl (Tuebingen University) for pointing out a vital issue in the earlier version of this paper, and the anonymous reviewers for rendering the current version clearer and more readable.

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  1. CNRS, Fédération de Mathématiques de CentraleSupélec FR 3487, Univ. Paris-Saclay, CentraleSupélec, 91190, Gif-sur-Yvette, France

    Jad Doghman

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This work is supported by a public grant as part of the Investissement d’avenir project [ANR-11-LABX-0056-LMH, LabEx LMH], and is part of the SIMALIN project [ANR-19-CE40-0016] of the French National Research Agency.

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Doghman, J. Numerical approximation of the stochastic Navier–Stokes equations through artificial compressibility. Calcolo 61, 23 (2024). https://doi.org/10.1007/s10092-024-00575-3

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