Optimal error estimates of the penalty finite element method for the unsteady Navier-Stokes equations with nonsmooth initial data
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866910130375753728 |
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| author | Bir, Bikram Goswami, Deepjyoti Pani, Amiya K. |
| author_facet | Bir, Bikram Goswami, Deepjyoti Pani, Amiya K. |
| contents | In this paper, both semidiscrete and fully discrete finite element methods are analyzed for the penalized two-dimensional unsteady Navier-Stokes equations with nonsmooth initial data. First order backward Euler method is applied for the time discretization, whereas conforming finite element method is used for the spatial discretization. Optimal $L^2$ error estimates for the semidiscrete as well as the fully discrete approximations of the velocity and of the pressure are derived for realistically assumed conditions on the data. The main ingredient in the proof is the appropriate exploitation of the inverse of the penalized Stokes operator, negative norm estimates and time weighted estimates. Numerical examples are discussed at the end which conform our theoretical results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_03777 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Optimal error estimates of the penalty finite element method for the unsteady Navier-Stokes equations with nonsmooth initial data Bir, Bikram Goswami, Deepjyoti Pani, Amiya K. Numerical Analysis 65M60, 65M15, 35Q35 In this paper, both semidiscrete and fully discrete finite element methods are analyzed for the penalized two-dimensional unsteady Navier-Stokes equations with nonsmooth initial data. First order backward Euler method is applied for the time discretization, whereas conforming finite element method is used for the spatial discretization. Optimal $L^2$ error estimates for the semidiscrete as well as the fully discrete approximations of the velocity and of the pressure are derived for realistically assumed conditions on the data. The main ingredient in the proof is the appropriate exploitation of the inverse of the penalized Stokes operator, negative norm estimates and time weighted estimates. Numerical examples are discussed at the end which conform our theoretical results. |
| title | Optimal error estimates of the penalty finite element method for the unsteady Navier-Stokes equations with nonsmooth initial data |
| topic | Numerical Analysis 65M60, 65M15, 35Q35 |
| url | https://arxiv.org/abs/2202.03777 |