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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.16917 |
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| _version_ | 1866915353119948800 |
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| author | García-Archilla, Bosco John, V. Novo, Julia |
| author_facet | García-Archilla, Bosco John, V. Novo, Julia |
| contents | Error bounds for fully discrete schemes for the evolutionary incompressible Navier--Stokes equations are derived in this paper. For the time integration we apply BDF-$q$ methods, $q\le 5$, for which error bounds for $q\ge 3$ cannot be found in the literature. Inf-sup stable mixed finite elements are used as spatial approximation. First, we analyze the standard Galerkin method and second a grad-div stabilized method. The grad-div stabilization allows to prove error bounds with constants independent of inverse powers of the viscosity coefficient. We prove optimal bounds for the velocity and pressure with order $(Δt)^q$ in time for the BDF-$q$ scheme and order $h^{k+1}$ for the $L^2(Ω)$ error of the velocity in the first case and $h^k$ in the second case, $k$ being the degree of the polynomials in finite element velocity space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_16917 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Error analysis of BDF schemes for the evolutionary incompressible Navier--Stokes equations García-Archilla, Bosco John, V. Novo, Julia Numerical Analysis Error bounds for fully discrete schemes for the evolutionary incompressible Navier--Stokes equations are derived in this paper. For the time integration we apply BDF-$q$ methods, $q\le 5$, for which error bounds for $q\ge 3$ cannot be found in the literature. Inf-sup stable mixed finite elements are used as spatial approximation. First, we analyze the standard Galerkin method and second a grad-div stabilized method. The grad-div stabilization allows to prove error bounds with constants independent of inverse powers of the viscosity coefficient. We prove optimal bounds for the velocity and pressure with order $(Δt)^q$ in time for the BDF-$q$ scheme and order $h^{k+1}$ for the $L^2(Ω)$ error of the velocity in the first case and $h^k$ in the second case, $k$ being the degree of the polynomials in finite element velocity space. |
| title | Error analysis of BDF schemes for the evolutionary incompressible Navier--Stokes equations |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2506.16917 |