Enhancing nonlinear solvers for the Navier-Stokes equations with continuous (noisy) data assimilation
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| Format: | Preprint |
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2024
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| _version_ | 1866911756334399488 |
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| author | Garcia-Archilla, Bosco Li, Xuejian Novo, Julia Rebholz, Leo |
| author_facet | Garcia-Archilla, Bosco Li, Xuejian Novo, Julia Rebholz, Leo |
| contents | We consider nonlinear solvers for the incompressible, steady (or at a fixed time step for unsteady) Navier-Stokes equations in the setting where partial measurement data of the solution is available. The measurement data is incorporated/assimilated into the solution through a nudging term addition to the the Picard iteration that penalized the difference between the coarse mesh interpolants of the true solution and solver solution, analogous to how continuous data assimilation (CDA) is implemented for time dependent PDEs. This was considered in the paper [Li et al. {\it CMAME} 2023], and we extend the methodology by improving the analysis to be in the $L^2$ norm instead of a weighted $H^1$ norm where the weight depended on the coarse mesh width, and to the case of noisy measurement data. For noisy measurement data, we prove that the CDA-Picard method is stable and convergent, up to the size of the noise. Numerical tests illustrate the results, and show that a very good strategy when using noisy data is to use CDA-Picard to generate an initial guess for the classical Newton iteration. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_06749 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Enhancing nonlinear solvers for the Navier-Stokes equations with continuous (noisy) data assimilation Garcia-Archilla, Bosco Li, Xuejian Novo, Julia Rebholz, Leo Numerical Analysis 65N12, 65N30 We consider nonlinear solvers for the incompressible, steady (or at a fixed time step for unsteady) Navier-Stokes equations in the setting where partial measurement data of the solution is available. The measurement data is incorporated/assimilated into the solution through a nudging term addition to the the Picard iteration that penalized the difference between the coarse mesh interpolants of the true solution and solver solution, analogous to how continuous data assimilation (CDA) is implemented for time dependent PDEs. This was considered in the paper [Li et al. {\it CMAME} 2023], and we extend the methodology by improving the analysis to be in the $L^2$ norm instead of a weighted $H^1$ norm where the weight depended on the coarse mesh width, and to the case of noisy measurement data. For noisy measurement data, we prove that the CDA-Picard method is stable and convergent, up to the size of the noise. Numerical tests illustrate the results, and show that a very good strategy when using noisy data is to use CDA-Picard to generate an initial guess for the classical Newton iteration. |
| title | Enhancing nonlinear solvers for the Navier-Stokes equations with continuous (noisy) data assimilation |
| topic | Numerical Analysis 65N12, 65N30 |
| url | https://arxiv.org/abs/2401.06749 |