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Analysis of Fully Discrete Mixed Finite Element Methods for Time-dependent Stochastic Stokes Equations with Multiplicative Noise

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Abstract

This paper is concerned with fully discrete mixed finite element approximations of the time-dependent stochastic Stokes equations with multiplicative noise. A prototypical method, which comprises of the Euler–Maruyama scheme for time discretization and the Taylor-Hood mixed element for spatial discretization is studied in detail. Strong convergence with rates is established not only for the velocity approximation but also for the pressure approximation (in a time-averaged fashion). A stochastic inf-sup condition is established and used in a nonstandard way to obtain the error estimate for the pressure approximation in the time-averaged fashion. Numerical results are also provided to validate the theoretical results and to gauge the performance of the proposed fully discrete mixed finite element methods.

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Acknowledgements

The authors would like to thank Professor Andreas Prohl of University of Tübingen (Germany) for his many stimulating discussions and critical comments as well as valuable suggestions which help to improve the early version of the paper considerably. In addition, his help on introducing and explaining several relevant references is also greatly appreciated.

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Authors and Affiliations

  1. Department of Mathematics, The University of Tennessee, Knoxville, TN, 37996, USA

    Xiaobing Feng

  2. School of Mathematics and Physics, Yancheng Institute of Technology, Yancheng, 224051, China

    Hailong Qiu

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  1. Xiaobing Feng
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  2. Hailong Qiu
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Correspondence to Xiaobing Feng.

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The work of the first author was partially supported by the NSF grants DMS-1620168 and DMS-2012414. The work of the second author was partially supported by the NSF of China grant 11701498.

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Feng, X., Qiu, H. Analysis of Fully Discrete Mixed Finite Element Methods for Time-dependent Stochastic Stokes Equations with Multiplicative Noise. J Sci Comput 88, 31 (2021). https://doi.org/10.1007/s10915-021-01546-4

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  • DOI: https://doi.org/10.1007/s10915-021-01546-4

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