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Constrained and unconstrained stable discrete minimizations for p-robust local reconstructions in vertex patches in the De Rham complex

Abstract : We analyze constrained and unconstrained minimization problems on patches of tetrahedra sharing a common vertex with discontinuous piecewise polynomial data of degree p. We show that the discrete minimizers in the spaces of piecewise polynomials of degree p conforming in the H1, H(curl), or H(div) spaces are as good as the minimizers in these entire (infinite-dimensional) Sobolev spaces, up to a constant that is independent of p. These results are useful in the analysis and design of finite element methods, namely for devising stable local commuting projectors and establishing local-best/global-best equivalences in a priori analysis and in the context of a posteriori error estimation. Unconstrained minimization in H1 and constrained minimization in H(div) have been previously treated in the literature. Along with improvement of the results in the H1 and H(div) cases, our key contribution is the treatment of the H(curl) framework. This enables us to cover the whole De Rham diagram in three space dimensions in a single setting.
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https://hal.inria.fr/hal-03749682
Contributeur : Théophile Chaumont-Frelet Connectez-vous pour contacter le contributeur
Soumis le : jeudi 11 août 2022 - 11:15:21
Dernière modification le : samedi 13 août 2022 - 12:13:58

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  • HAL Id : hal-03749682, version 1
  • ARXIV : 2208.05870

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Théophile Chaumont-Frelet, Martin Vohralík. Constrained and unconstrained stable discrete minimizations for p-robust local reconstructions in vertex patches in the De Rham complex. 2022. ⟨hal-03749682⟩

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