Locally-Verifiable Sufficient Conditions for Exactness of the Hierarchical B-spline Discrete de Rham Complex in Rn.
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| Title: | Locally-Verifiable Sufficient Conditions for Exactness of the Hierarchical B-spline Discrete de Rham Complex in Rn. |
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| Authors: | Shepherd, Kendrick1 (AUTHOR) kendrick_shepherd@byu.edu, Toshniwal, Deepesh2 (AUTHOR) d.toshniwal@tudelft.nl |
| Source: | Foundations of Computational Mathematics. Feb2026, Vol. 26 Issue 1, p525-567. 43p. |
| Subjects: | Differential forms, Spline theory, Discrete exterior calculus, Algebraic topology, Numerical analysis |
| Abstract: | Given a domain Ω ⊂ R n , the de Rham complex of differential forms arises naturally in the study of problems in electromagnetism and fluid mechanics defined on Ω , and its discretization helps build stable numerical methods for such problems. For constructing such stable methods, one critical requirement is ensuring that the discrete subcomplex is cohomologically equivalent to the continuous complex. When Ω is a hypercube, we thus require that the discrete subcomplex be exact. Focusing on such Ω , we theoretically analyze the discrete de Rham complex built from hierarchical B-spline differential forms, i.e., the discrete differential forms are smooth splines and support adaptive refinements—these properties are key to enabling accurate and efficient numerical simulations. We provide locally-verifiable sufficient conditions that ensure that the discrete spline complex is exact. Numerical tests are presented to support the theoretical results, and the examples discussed include complexes that satisfy our prescribed conditions as well as those that violate them. [ABSTRACT FROM AUTHOR] |
| Copyright of Foundations of Computational Mathematics is the property of Springer Nature 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 |
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| Header | DbId: egs DbLabel: Engineering Source An: 192095335 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Locally-Verifiable Sufficient Conditions for Exactness of the Hierarchical B-spline Discrete de Rham Complex in Rn. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Shepherd%2C+Kendrick%22">Shepherd, Kendrick</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> kendrick_shepherd@byu.edu</i><br /><searchLink fieldCode="AR" term="%22Toshniwal%2C+Deepesh%22">Toshniwal, Deepesh</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> d.toshniwal@tudelft.nl</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Foundations+of+Computational+Mathematics%22">Foundations of Computational Mathematics</searchLink>. Feb2026, Vol. 26 Issue 1, p525-567. 43p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Differential+forms%22">Differential forms</searchLink><br /><searchLink fieldCode="DE" term="%22Spline+theory%22">Spline theory</searchLink><br /><searchLink fieldCode="DE" term="%22Discrete+exterior+calculus%22">Discrete exterior calculus</searchLink><br /><searchLink fieldCode="DE" term="%22Algebraic+topology%22">Algebraic topology</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+analysis%22">Numerical analysis</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: Given a domain Ω ⊂ R n , the de Rham complex of differential forms arises naturally in the study of problems in electromagnetism and fluid mechanics defined on Ω , and its discretization helps build stable numerical methods for such problems. For constructing such stable methods, one critical requirement is ensuring that the discrete subcomplex is cohomologically equivalent to the continuous complex. When Ω is a hypercube, we thus require that the discrete subcomplex be exact. Focusing on such Ω , we theoretically analyze the discrete de Rham complex built from hierarchical B-spline differential forms, i.e., the discrete differential forms are smooth splines and support adaptive refinements—these properties are key to enabling accurate and efficient numerical simulations. We provide locally-verifiable sufficient conditions that ensure that the discrete spline complex is exact. Numerical tests are presented to support the theoretical results, and the examples discussed include complexes that satisfy our prescribed conditions as well as those that violate them. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Foundations of Computational Mathematics is the property of Springer Nature 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: 10.1007/s10208-024-09659-6 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 43 StartPage: 525 Subjects: – SubjectFull: Differential forms Type: general – SubjectFull: Spline theory Type: general – SubjectFull: Discrete exterior calculus Type: general – SubjectFull: Algebraic topology Type: general – SubjectFull: Numerical analysis Type: general Titles: – TitleFull: Locally-Verifiable Sufficient Conditions for Exactness of the Hierarchical B-spline Discrete de Rham Complex in Rn. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Shepherd, Kendrick – PersonEntity: Name: NameFull: Toshniwal, Deepesh IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 02 Text: Feb2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 16153375 Numbering: – Type: volume Value: 26 – Type: issue Value: 1 Titles: – TitleFull: Foundations of Computational Mathematics Type: main |
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