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Generalized Gaffney inequality and discrete compactness for discrete differential forms

Abstract

We prove generalized Gaffney inequalities and the discrete compactness for finite element differential forms on s-regular domains, including general Lipschitz domains. In computational electromagnetism, special cases of these results have been established for edge elements with weakly imposed divergence-free conditions and used in the analysis of nonlinear and eigenvalue problems. In this paper, we generalize these results to discrete differential forms, not necessarily with strongly or weakly imposed constraints. The analysis relies on a new Hodge mapping and its approximation property. As an application, we show \(L^{p}\) estimates for several finite element approximations of the scalar and vector Laplacian problems.

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Acknowledgements

The authors are grateful to Prof. Ralf Hiptmair for several helpful suggestions.

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Correspondence to Kaibo Hu.

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Juncai He is supported in part by National Natural Science Foundation of China (NSFC) (Grant No. 91430215) and the Elite Program of Computational and Applied Mathematics for PHD Candidates of Peking University. The research of Kaibo Hu leading to the results of this paper was partly carried out during his affiliation with the University of Oslo. KH was then supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC Grant Agreement 339643 (FEEC-A). Jinchao Xu is supported in part by DOE Grant DE-SC0014400.

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He, J., Hu, K. & Xu, J. Generalized Gaffney inequality and discrete compactness for discrete differential forms. Numer. Math. 143, 781–795 (2019). https://doi.org/10.1007/s00211-019-01076-0

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Mathematics Subject Classification

  • 65N30
  • 65N12
  • 58J10
  • 35D30