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Volume 36, Issue 3
High Moment and Pathwise Error Estimates for Fully Discrete Mixed Finite Element Approximations of Stochastic Navier-Stokes Equations with Additive Noise

Xiaobing Feng & Liet Vo

DOI: 10.4208/cicp.OA-2023-0234

Commun. Comput. Phys., 36 (2024), pp. 821-849.

Published online: 2024-10

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  • Abstract

This paper is concerned with high moment and pathwise error estimates for fully discrete mixed finite element approximations of stochastic Navier-Stokes equations with general additive noise. The implicit Euler-Maruyama scheme and standard mixed finite element methods are employed respectively for the time and space discretizations. High moment error estimates for both velocity and time-averaged pressure approximations in strong $L^2$ and energy norms are obtained, pathwise error estimates are derived by using the Kolmogorov Theorem. Unlike their deterministic counterparts, the spatial error constants grow in the order of $\mathcal{O}(k^{-\frac{1}{2}} ),$ where $k$ denotes time step size. Numerical experiments are also provided to validate the error estimates and their sharpness.

  • Keywords

Stochastic Navier-Stokes equations, additive noise, Wiener process, Itô stochastic integral, mixed finite element methods, inf-sup condition, high moment, and pathwise error estimates.

  • AMS Subject Headings

65N12, 65N15, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-36-821, author = {Feng , Xiaobing and Vo , Liet}, title = {High Moment and Pathwise Error Estimates for Fully Discrete Mixed Finite Element Approximations of Stochastic Navier-Stokes Equations with Additive Noise}, journal = {Communications in Computational Physics}, year = {2024}, volume = {36}, number = {3}, pages = {821--849}, abstract = {

This paper is concerned with high moment and pathwise error estimates for fully discrete mixed finite element approximations of stochastic Navier-Stokes equations with general additive noise. The implicit Euler-Maruyama scheme and standard mixed finite element methods are employed respectively for the time and space discretizations. High moment error estimates for both velocity and time-averaged pressure approximations in strong $L^2$ and energy norms are obtained, pathwise error estimates are derived by using the Kolmogorov Theorem. Unlike their deterministic counterparts, the spatial error constants grow in the order of $\mathcal{O}(k^{-\frac{1}{2}} ),$ where $k$ denotes time step size. Numerical experiments are also provided to validate the error estimates and their sharpness.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2023-0234}, url = {http://global-sci.org/intro/article_detail/cicp/23459.html} }
TY - JOUR T1 - High Moment and Pathwise Error Estimates for Fully Discrete Mixed Finite Element Approximations of Stochastic Navier-Stokes Equations with Additive Noise AU - Feng , Xiaobing AU - Vo , Liet JO - Communications in Computational Physics VL - 3 SP - 821 EP - 849 PY - 2024 DA - 2024/10 SN - 36 DO - http://doi.org/10.4208/cicp.OA-2023-0234 UR - https://global-sci.org/intro/article_detail/cicp/23459.html KW - Stochastic Navier-Stokes equations, additive noise, Wiener process, Itô stochastic integral, mixed finite element methods, inf-sup condition, high moment, and pathwise error estimates. AB -

This paper is concerned with high moment and pathwise error estimates for fully discrete mixed finite element approximations of stochastic Navier-Stokes equations with general additive noise. The implicit Euler-Maruyama scheme and standard mixed finite element methods are employed respectively for the time and space discretizations. High moment error estimates for both velocity and time-averaged pressure approximations in strong $L^2$ and energy norms are obtained, pathwise error estimates are derived by using the Kolmogorov Theorem. Unlike their deterministic counterparts, the spatial error constants grow in the order of $\mathcal{O}(k^{-\frac{1}{2}} ),$ where $k$ denotes time step size. Numerical experiments are also provided to validate the error estimates and their sharpness.

Xiaobing Feng & Liet Vo. (2024). High Moment and Pathwise Error Estimates for Fully Discrete Mixed Finite Element Approximations of Stochastic Navier-Stokes Equations with Additive Noise. Communications in Computational Physics. 36 (3). 821-849. doi:10.4208/cicp.OA-2023-0234
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