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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| Online-Zugang: | https://arxiv.org/abs/2210.15572 |
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| _version_ | 1866915090812370944 |
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| author | Ko, Seungchan Li, Guanglian Yu, Yi |
| author_facet | Ko, Seungchan Li, Guanglian Yu, Yi |
| contents | In this paper, we analyze the numerical approximation of the Navier-Stokes problem over a bounded polygonal domain in $\mathbb{R}^2$, where the initial condition is modeled by a log-normal random field. This problem usually arises in the area of uncertainty quantification. We aim to compute the expectation value of linear functionals of the solution to the Navier-Stokes equations and perform a rigorous error analysis for the problem. In particular, our method includes the finite element, fully-discrete discretizations, truncated Karhunen-Loéve expansion for the realizations of the initial condition, and lattice-based quasi-Monte Carlo (QMC) method to estimate the expected values over the parameter space. Our QMC analysis is based on randomly-shifted lattice rules for the integration over the domain in high-dimensional space, which guarantees the error decays with $\mathcal{O}(N^{-1+δ})$, where $N$ is the number of sampling points, $δ>0$ is an arbitrary small number, and the constant in the decay estimate is independent of the dimension of integration. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_15572 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Quasi-Monte Carlo finite element approximation of the Navier-Stokes equations with initial data modeled by log-normal random fields Ko, Seungchan Li, Guanglian Yu, Yi Numerical Analysis In this paper, we analyze the numerical approximation of the Navier-Stokes problem over a bounded polygonal domain in $\mathbb{R}^2$, where the initial condition is modeled by a log-normal random field. This problem usually arises in the area of uncertainty quantification. We aim to compute the expectation value of linear functionals of the solution to the Navier-Stokes equations and perform a rigorous error analysis for the problem. In particular, our method includes the finite element, fully-discrete discretizations, truncated Karhunen-Loéve expansion for the realizations of the initial condition, and lattice-based quasi-Monte Carlo (QMC) method to estimate the expected values over the parameter space. Our QMC analysis is based on randomly-shifted lattice rules for the integration over the domain in high-dimensional space, which guarantees the error decays with $\mathcal{O}(N^{-1+δ})$, where $N$ is the number of sampling points, $δ>0$ is an arbitrary small number, and the constant in the decay estimate is independent of the dimension of integration. |
| title | Quasi-Monte Carlo finite element approximation of the Navier-Stokes equations with initial data modeled by log-normal random fields |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2210.15572 |