A Unified Framework for Multiscale Spectral Generalized FEMs and Low-Rank Approximations to Multiscale PDEs.
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| Title: | A Unified Framework for Multiscale Spectral Generalized FEMs and Low-Rank Approximations to Multiscale PDEs. |
|---|---|
| Authors: | Ma, Chupeng1 (AUTHOR) chupeng.ma@gbu.edu.cn |
| Source: | Foundations of Computational Mathematics. Jun2026, Vol. 26 Issue 3, p1699-1758. 60p. |
| Subjects: | Multiscale modeling, Low-rank matrices, Green's functions, Numerical analysis, Finite element method, Spectral element method, Approximation theory, Computational mathematics |
| Abstract: | Multiscale partial differential equations (PDEs), featuring heterogeneous coefficients oscillating across possibly non-separated scales, pose computational challenges for standard numerical techniques. Over the past two decades, a range of specialized methods has emerged that enables the efficient solution of such problems. Two prominent approaches are numerical multiscale methods with problem-adapted coarse approximation spaces, and structured inverse methods that exploit a low-rank property of the associated Green's functions to obtain approximate matrix factorizations. This work presents an abstract framework for the design, implementation, and analysis of the multiscale spectral generalized finite element method (MS-GFEM), a particular numerical multiscale method originally proposed in Babuska and Lipton (Multiscale Model Simul 9:373–406, 2011). MS-GFEM is a partition of unity method employing optimal local approximation spaces constructed from local spectral problems. We establish a general local approximation theory demonstrating exponential convergence with respect to the number of local degrees of freedom under certain assumptions, with explicit dependence on key problem parameters. Our framework applies to a broad class of multiscale PDEs with L ∞ -coefficients in both continuous and discrete, finite element settings, including highly indefinite problems and higher-order problems. Notably, we prove a local convergence rate of O (e - c n 1 / d ) for MS-GFEM for all these problems, improving upon the O (e - c n 1 / (d + 1) ) rate shown by Babuska and Lipton. Moreover, based on the abstract local approximation theory for MS-GFEM, we establish a unified framework for showing low-rank approximations to multiscale PDEs. This framework applies to the aforementioned problems, proving that the associated Green's functions admit an O (| log ϵ | d) -term separable approximation on well-separated domains with error ϵ > 0 . Our analysis improves and generalizes the result in Bebendorf and Hackbusch (Numerische Mathematik 95:1–28, 2003) where an O (| log ϵ | d + 1) -term separable approximation was proved for Poisson-type problems. It provides a rigorous theoretical foundation for diverse structured inverse methods, and also clarifies the intimate connection between approximation mechanisms in such methods and MS-GFEM. [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: 194201087 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: A Unified Framework for Multiscale Spectral Generalized FEMs and Low-Rank Approximations to Multiscale PDEs. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Ma%2C+Chupeng%22">Ma, Chupeng</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> chupeng.ma@gbu.edu.cn</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Foundations+of+Computational+Mathematics%22">Foundations of Computational Mathematics</searchLink>. Jun2026, Vol. 26 Issue 3, p1699-1758. 60p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Multiscale+modeling%22">Multiscale modeling</searchLink><br /><searchLink fieldCode="DE" term="%22Low-rank+matrices%22">Low-rank matrices</searchLink><br /><searchLink fieldCode="DE" term="%22Green's+functions%22">Green's functions</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+analysis%22">Numerical analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Finite+element+method%22">Finite element method</searchLink><br /><searchLink fieldCode="DE" term="%22Spectral+element+method%22">Spectral element method</searchLink><br /><searchLink fieldCode="DE" term="%22Approximation+theory%22">Approximation theory</searchLink><br /><searchLink fieldCode="DE" term="%22Computational+mathematics%22">Computational mathematics</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: Multiscale partial differential equations (PDEs), featuring heterogeneous coefficients oscillating across possibly non-separated scales, pose computational challenges for standard numerical techniques. Over the past two decades, a range of specialized methods has emerged that enables the efficient solution of such problems. Two prominent approaches are numerical multiscale methods with problem-adapted coarse approximation spaces, and structured inverse methods that exploit a low-rank property of the associated Green's functions to obtain approximate matrix factorizations. This work presents an abstract framework for the design, implementation, and analysis of the multiscale spectral generalized finite element method (MS-GFEM), a particular numerical multiscale method originally proposed in Babuska and Lipton (Multiscale Model Simul 9:373–406, 2011). MS-GFEM is a partition of unity method employing optimal local approximation spaces constructed from local spectral problems. We establish a general local approximation theory demonstrating exponential convergence with respect to the number of local degrees of freedom under certain assumptions, with explicit dependence on key problem parameters. Our framework applies to a broad class of multiscale PDEs with L ∞ -coefficients in both continuous and discrete, finite element settings, including highly indefinite problems and higher-order problems. Notably, we prove a local convergence rate of O (e - c n 1 / d ) for MS-GFEM for all these problems, improving upon the O (e - c n 1 / (d + 1) ) rate shown by Babuska and Lipton. Moreover, based on the abstract local approximation theory for MS-GFEM, we establish a unified framework for showing low-rank approximations to multiscale PDEs. This framework applies to the aforementioned problems, proving that the associated Green's functions admit an O (| log ϵ | d) -term separable approximation on well-separated domains with error ϵ > 0 . Our analysis improves and generalizes the result in Bebendorf and Hackbusch (Numerische Mathematik 95:1–28, 2003) where an O (| log ϵ | d + 1) -term separable approximation was proved for Poisson-type problems. It provides a rigorous theoretical foundation for diverse structured inverse methods, and also clarifies the intimate connection between approximation mechanisms in such methods and MS-GFEM. [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/s10208-025-09711-z Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 60 StartPage: 1699 Subjects: – SubjectFull: Multiscale modeling Type: general – SubjectFull: Low-rank matrices Type: general – SubjectFull: Green's functions Type: general – SubjectFull: Numerical analysis Type: general – SubjectFull: Finite element method Type: general – SubjectFull: Spectral element method Type: general – SubjectFull: Approximation theory Type: general – SubjectFull: Computational mathematics Type: general Titles: – TitleFull: A Unified Framework for Multiscale Spectral Generalized FEMs and Low-Rank Approximations to Multiscale PDEs. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Ma, Chupeng IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 06 Text: Jun2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 16153375 Numbering: – Type: volume Value: 26 – Type: issue Value: 3 Titles: – TitleFull: Foundations of Computational Mathematics Type: main |
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