<div class="csl-bib-body">
<div class="csl-entry">Bringmann, P., Ketteler, J., & Schedensack, M. (2024). Discrete Helmholtz Decompositions of Piecewise Constant and Piecewise Affine Vector and Tensor Fields. <i>Foundations of Computational Mathematics</i>. https://doi.org/10.1007/s10208-024-09642-1</div>
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dc.identifier.issn
1615-3375
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dc.identifier.uri
http://hdl.handle.net/20.500.12708/208761
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dc.description.abstract
Discrete Helmholtz decompositions dissect piecewise polynomial vector fields on simplicial meshes into piecewise gradients and rotations of finite element functions. This paper concisely reviews established results from the literature which all restrict to the lowest-order case of piecewise constants. Its main contribution consists of the generalization of these decompositions to 3D and of novel decompositions for piecewise affine vector fields in terms of Fortin–Soulie functions. While the classical lowest-order decompositions include one conforming and one nonconforming part, the decompositions of piecewise affine vector fields require a nonconforming enrichment in both parts. The presentation covers two and three spatial dimensions as well as generalizations to deviatoric tensor fields in the context of the Stokes equations and symmetric tensor fields for the linear elasticity and fourth-order problems. While the proofs focus on contractible domains, generalizations to multiply connected domains and domains with non-connected boundary are discussed as well.
en
dc.language.iso
en
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dc.publisher
SPRINGER
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dc.relation.ispartof
Foundations of Computational Mathematics
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dc.rights.uri
http://creativecommons.org/licenses/by/4.0/
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dc.subject
Crouzeix–Raviart
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dc.subject
Discrete Helmholtz decompositions
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dc.subject
Fortin–Soulie
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dc.subject
Fourth-order problems
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dc.subject
Linear elasticity
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dc.subject
Mixed FEM
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dc.subject
Nonconforming FEM
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dc.subject
Stokes equations
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dc.title
Discrete Helmholtz Decompositions of Piecewise Constant and Piecewise Affine Vector and Tensor Fields