Error estimates for finite element discretizations of the instationary Navier-Stokes equations
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911697437982720 |
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| author | Vexler, Boris Wagner, Jakob |
| author_facet | Vexler, Boris Wagner, Jakob |
| contents | In this work we consider the two dimensional instationary Navier-Stokes equations with homogeneous Dirichlet/no-slip boundary conditions. We show error estimates for the fully discrete problem, where a discontinuous Galerkin method in time and inf-sup stable finite elements in space are used. Recently, best approximation type error estimates for the Stokes problem in the $L^\infty(I;L^2(Ω))$, $L^2(I;H^1(Ω))$ and $L^2(I;L^2(Ω))$ norms have been shown. The main result of the present work extends the error estimate in the $L^\infty(I;L^2(Ω))$ norm to the Navier-Stokes equations, by pursuing an error splitting approach and an appropriate duality argument. In order to discuss the stability of solutions to the discrete primal and dual equations, a specially tailored discrete Gronwall lemma is presented. The techniques developed towards showing the $L^\infty(I;L^2(Ω))$ error estimate, also allow us to show best approximation type error estimates in the $L^2(I;H^1(Ω))$ and $L^2(I;L^2(Ω))$ norms, which complement this work. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_14217 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Error estimates for finite element discretizations of the instationary Navier-Stokes equations Vexler, Boris Wagner, Jakob Numerical Analysis 35Q30, 65M60, 65M15, 65M22, 76D05, 76M10 In this work we consider the two dimensional instationary Navier-Stokes equations with homogeneous Dirichlet/no-slip boundary conditions. We show error estimates for the fully discrete problem, where a discontinuous Galerkin method in time and inf-sup stable finite elements in space are used. Recently, best approximation type error estimates for the Stokes problem in the $L^\infty(I;L^2(Ω))$, $L^2(I;H^1(Ω))$ and $L^2(I;L^2(Ω))$ norms have been shown. The main result of the present work extends the error estimate in the $L^\infty(I;L^2(Ω))$ norm to the Navier-Stokes equations, by pursuing an error splitting approach and an appropriate duality argument. In order to discuss the stability of solutions to the discrete primal and dual equations, a specially tailored discrete Gronwall lemma is presented. The techniques developed towards showing the $L^\infty(I;L^2(Ω))$ error estimate, also allow us to show best approximation type error estimates in the $L^2(I;H^1(Ω))$ and $L^2(I;L^2(Ω))$ norms, which complement this work. |
| title | Error estimates for finite element discretizations of the instationary Navier-Stokes equations |
| topic | Numerical Analysis 35Q30, 65M60, 65M15, 65M22, 76D05, 76M10 |
| url | https://arxiv.org/abs/2307.14217 |