Mixed finite element methods for linear Cosserat equations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910656999981056 |
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| author | Boon, Wietse Marijn Duran, Omar Nordbotten, Jan Martin |
| author_facet | Boon, Wietse Marijn Duran, Omar Nordbotten, Jan Martin |
| contents | We consider the equilibrium equations for a linearized Cosserat material and provide two perspectives concerning well-posedness. First, the system can be viewed as the Hodge Laplace problem on a differential complex. On the other hand, we show how the Cosserat materials can be analyzed by inheriting results from linearized elasticity. Both perspectives give rise to mixed finite element methods, which we refer to as strongly and weakly coupled, respectively. We prove convergence of both classes of methods, with particular attention to improved convergence rate estimates, and stability in the limit of vanishing characteristic length of the micropolar structure. The theoretical results are fully reflected in the actual performance of the methods, as shown by the numerical verifications. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_15136 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mixed finite element methods for linear Cosserat equations Boon, Wietse Marijn Duran, Omar Nordbotten, Jan Martin Numerical Analysis 68Q25, 68R10, 68U05 We consider the equilibrium equations for a linearized Cosserat material and provide two perspectives concerning well-posedness. First, the system can be viewed as the Hodge Laplace problem on a differential complex. On the other hand, we show how the Cosserat materials can be analyzed by inheriting results from linearized elasticity. Both perspectives give rise to mixed finite element methods, which we refer to as strongly and weakly coupled, respectively. We prove convergence of both classes of methods, with particular attention to improved convergence rate estimates, and stability in the limit of vanishing characteristic length of the micropolar structure. The theoretical results are fully reflected in the actual performance of the methods, as shown by the numerical verifications. |
| title | Mixed finite element methods for linear Cosserat equations |
| topic | Numerical Analysis 68Q25, 68R10, 68U05 |
| url | https://arxiv.org/abs/2403.15136 |