Polytopal templates for semi-continuous vectorial finite elements of arbitrary order on triangulations and tetrahedralizations.
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| Title: | Polytopal templates for semi-continuous vectorial finite elements of arbitrary order on triangulations and tetrahedralizations. |
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| Authors: | Sky, Adam1 (AUTHOR) adam.sky@tu-dortmund.de, Muench, Ingo1 (AUTHOR) ingo.muench@tu-dortmund.de |
| Source: | Finite Elements in Analysis & Design. Sep2024, Vol. 236, pN.PAG-N.PAG. 1p. |
| Subjects: | Hilbert space, Triangulation, Polytopes, Triangles, Polynomials |
| Abstract: | The Hilbert spaces H (curl) and H (div) are employed in various variational problems formulated in the context of the de Rham complex in order to guarantee well-posedness. Seeing as the well-posedness follows automatically from the continuous setting to the discrete setting in the presence of commuting interpolants as per Fortin's criterion, the construction of conforming subspaces becomes a crucial step in the formulation of stable numerical schemes. This work aims to introduce a novel, simple method of directly constructing semi-continuous vectorial base functions on the reference element via template vectors associated with the geometric polytopes of the element and an underlying H 1 -conforming polynomial subspace. The base functions are then mapped from the reference element to the element in the physical domain via consistent Piola transformations. The method is defined in such a way, that the underlying H 1 -conforming subspace can be chosen independently, thus allowing for constructions of arbitrary polynomial order. We prove a linearly independent construction of Nédélec elements of the first and second type, Brezzi–Douglas–Marini elements, and Raviart–Thomas elements on triangulations and tetrahedralizations. The application of the method is demonstrated with two examples in the relaxed micromorphic model. • A new method of constructing semi-continuous vectorial finite elements via polytopes. • Proofs of linear independence and conformity for hierarchical- and partition of unity-bases. • Templates for Nédélec elements of the first and second type on simplices. • Templates for Brezzi–Douglas–Marini and Raviart–Thomas elements on simplices. • Split in the basis between kernel and non-kernel base functions on triangles. [ABSTRACT FROM AUTHOR] |
| Copyright of Finite Elements in Analysis & Design is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Text: Availability: 0 |
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| Header | DbId: egs DbLabel: Engineering Source An: 177420110 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Polytopal templates for semi-continuous vectorial finite elements of arbitrary order on triangulations and tetrahedralizations. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Sky%2C+Adam%22">Sky, Adam</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> adam.sky@tu-dortmund.de</i><br /><searchLink fieldCode="AR" term="%22Muench%2C+Ingo%22">Muench, Ingo</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> ingo.muench@tu-dortmund.de</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Finite+Elements+in+Analysis+%26+Design%22">Finite Elements in Analysis & Design</searchLink>. Sep2024, Vol. 236, pN.PAG-N.PAG. 1p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Hilbert+space%22">Hilbert space</searchLink><br /><searchLink fieldCode="DE" term="%22Triangulation%22">Triangulation</searchLink><br /><searchLink fieldCode="DE" term="%22Polytopes%22">Polytopes</searchLink><br /><searchLink fieldCode="DE" term="%22Triangles%22">Triangles</searchLink><br /><searchLink fieldCode="DE" term="%22Polynomials%22">Polynomials</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: The Hilbert spaces H (curl) and H (div) are employed in various variational problems formulated in the context of the de Rham complex in order to guarantee well-posedness. Seeing as the well-posedness follows automatically from the continuous setting to the discrete setting in the presence of commuting interpolants as per Fortin's criterion, the construction of conforming subspaces becomes a crucial step in the formulation of stable numerical schemes. This work aims to introduce a novel, simple method of directly constructing semi-continuous vectorial base functions on the reference element via template vectors associated with the geometric polytopes of the element and an underlying H 1 -conforming polynomial subspace. The base functions are then mapped from the reference element to the element in the physical domain via consistent Piola transformations. The method is defined in such a way, that the underlying H 1 -conforming subspace can be chosen independently, thus allowing for constructions of arbitrary polynomial order. We prove a linearly independent construction of Nédélec elements of the first and second type, Brezzi–Douglas–Marini elements, and Raviart–Thomas elements on triangulations and tetrahedralizations. The application of the method is demonstrated with two examples in the relaxed micromorphic model. • A new method of constructing semi-continuous vectorial finite elements via polytopes. • Proofs of linear independence and conformity for hierarchical- and partition of unity-bases. • Templates for Nédélec elements of the first and second type on simplices. • Templates for Brezzi–Douglas–Marini and Raviart–Thomas elements on simplices. • Split in the basis between kernel and non-kernel base functions on triangles. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Finite Elements in Analysis & Design is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1016/j.finel.2024.104155 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 1 StartPage: N.PAG Subjects: – SubjectFull: Hilbert space Type: general – SubjectFull: Triangulation Type: general – SubjectFull: Polytopes Type: general – SubjectFull: Triangles Type: general – SubjectFull: Polynomials Type: general Titles: – TitleFull: Polytopal templates for semi-continuous vectorial finite elements of arbitrary order on triangulations and tetrahedralizations. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Sky, Adam – PersonEntity: Name: NameFull: Muench, Ingo IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 09 Text: Sep2024 Type: published Y: 2024 Identifiers: – Type: issn-print Value: 0168874X Numbering: – Type: volume Value: 236 Titles: – TitleFull: Finite Elements in Analysis & Design Type: main |
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