Variational problems with gradient constraints: $\textit{A priori}$ and $\textit{a posteriori}$ error identities
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929557242642432 |
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| author | Antil, Harbir Bartels, Sören Kaltenbach, Alex Khandelwal, Rohit |
| author_facet | Antil, Harbir Bartels, Sören Kaltenbach, Alex Khandelwal, Rohit |
| contents | In this paper, on the basis of a (Fenchel) duality theory on the continuous level, we derive an $\textit{a posteriori}$ error identity for arbitrary conforming approximations of a primal formulation and a dual formulation of variational problems involving gradient constraints. In addition, on the basis of a (Fenchel) duality theory on the discrete level, we derive an $\textit{a priori}$ error identity that applies to the approximation of the primal formulation using the Crouzeix-Raviart element and to the approximation of the dual formulation using the Raviart-Thomas element, and leads to error decay rates that are optimal with respect to the regularity of a dual solution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_18780 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Variational problems with gradient constraints: $\textit{A priori}$ and $\textit{a posteriori}$ error identities Antil, Harbir Bartels, Sören Kaltenbach, Alex Khandelwal, Rohit Numerical Analysis 49J40, 49M29, 65N30, 65N15, 65N50 In this paper, on the basis of a (Fenchel) duality theory on the continuous level, we derive an $\textit{a posteriori}$ error identity for arbitrary conforming approximations of a primal formulation and a dual formulation of variational problems involving gradient constraints. In addition, on the basis of a (Fenchel) duality theory on the discrete level, we derive an $\textit{a priori}$ error identity that applies to the approximation of the primal formulation using the Crouzeix-Raviart element and to the approximation of the dual formulation using the Raviart-Thomas element, and leads to error decay rates that are optimal with respect to the regularity of a dual solution. |
| title | Variational problems with gradient constraints: $\textit{A priori}$ and $\textit{a posteriori}$ error identities |
| topic | Numerical Analysis 49J40, 49M29, 65N30, 65N15, 65N50 |
| url | https://arxiv.org/abs/2410.18780 |