Vector‐Valued Nonconforming Finite Element for Surface Flows
ABSTRACT In the recent years, there has been an increasing interest in the analysis of finite element methods for vector‐valued flow problems on curved geometries. In this contribution, we derive an error analysis for a vector‐valued Crouzeix–Raviart element. The derivation is performed on the vecto...
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| Published in | Proceedings in applied mathematics and mechanics Vol. 25; no. 1 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
01.03.2025
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| ISSN | 1617-7061 1617-7061 |
| DOI | 10.1002/pamm.202400116 |
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| Abstract | ABSTRACT
In the recent years, there has been an increasing interest in the analysis of finite element methods for vector‐valued flow problems on curved geometries. In this contribution, we derive an error analysis for a vector‐valued Crouzeix–Raviart element. The derivation is performed on the vector‐valued Laplace problem, which includes the symmetrical strain rate tensor, an important operator for modeling flow problems. The approximation of the strain rate tensor with the Crouzeix–Raviart element leads to oscillations in the velocity field due to a nontrivial kernel. We derive a stabilized form of the equation and present optimal error bounds in the H1$H^1$‐norm for the Crouzeix–Raviart finite element. The theoretical findings are supported by numerical results. |
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| AbstractList | In the recent years, there has been an increasing interest in the analysis of finite element methods for vector‐valued flow problems on curved geometries. In this contribution, we derive an error analysis for a vector‐valued Crouzeix–Raviart element. The derivation is performed on the vector‐valued Laplace problem, which includes the symmetrical strain rate tensor, an important operator for modeling flow problems. The approximation of the strain rate tensor with the Crouzeix–Raviart element leads to oscillations in the velocity field due to a nontrivial kernel. We derive a stabilized form of the equation and present optimal error bounds in the ‐norm for the Crouzeix–Raviart finite element. The theoretical findings are supported by numerical results. ABSTRACT In the recent years, there has been an increasing interest in the analysis of finite element methods for vector‐valued flow problems on curved geometries. In this contribution, we derive an error analysis for a vector‐valued Crouzeix–Raviart element. The derivation is performed on the vector‐valued Laplace problem, which includes the symmetrical strain rate tensor, an important operator for modeling flow problems. The approximation of the strain rate tensor with the Crouzeix–Raviart element leads to oscillations in the velocity field due to a nontrivial kernel. We derive a stabilized form of the equation and present optimal error bounds in the H1$H^1$‐norm for the Crouzeix–Raviart finite element. The theoretical findings are supported by numerical results. |
| Author | Mehlmann, Carolin |
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| Cites_doi | 10.1137/21M1403126 10.4171/ifb/405 10.1090/mcom/3551 10.1002/pamm.202300207 10.1137/17M1146038 10.1002/nme.6317 10.1137/18M1166183 10.1016/bs.hna.2019.06.002 10.1051/m2an/197307R300331 10.1007/s00211‐015‐0787‐5 10.1515/jnma-2020-0017 10.1137/18M1176464 10.1051/m2an:2003020 10.1090/mcom/3179 10.1017/jfm.2012.317 10.1137/19M1284592 10.1007/s10208‐012‐9119‐7 10.1137/23M1583995 10.1093/imanum/drz018 10.1029/2022MS003010 |
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In the recent years, there has been an increasing interest in the analysis of finite element methods for vector‐valued flow problems on curved... In the recent years, there has been an increasing interest in the analysis of finite element methods for vector‐valued flow problems on curved geometries. In... |
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| Title | Vector‐Valued Nonconforming Finite Element for Surface Flows |
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