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

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

Year:    2024

Author:    Xiaobing Feng, Liet Vo

Communications in Computational Physics, Vol. 36 (2024), Iss. 3 : pp. 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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cicp.OA-2023-0234

Communications in Computational Physics, Vol. 36 (2024), Iss. 3 : pp. 821–849

Published online:    2024-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    29

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.

Author Details

Xiaobing Feng

Liet Vo