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SPECTRAL ACMS: A ROBUST LOCALIZED APPROXIMATED COMPONENT MODE SYNTHESIS METHOD.

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Title: SPECTRAL ACMS: A ROBUST LOCALIZED APPROXIMATED COMPONENT MODE SYNTHESIS METHOD.
Authors: MADUREIRA, ALEXANDRE L.1 alexandre.madureira@fgv.br, SARKIS, MARCUS2 msarkis@wpi.edu
Source: SIAM Journal on Numerical Analysis. 2025, Vol. 63 Issue 3, p1055-1077. 23p.
Subjects: Finite element method, Elliptic differential equations, Approximation error, Multiple scale method, Reduced-order models, Galerkin methods
Abstract: We consider finite element methods of multiscale type to approximate solutions for two-dimensional symmetric elliptic partial differential equations with heterogeneous L∞ coefficients. The methods are of Galerkin type and follow the Variational Multiscale and Localized Orthogonal Decomposition (LOD) approaches in the sense that it decouples spaces into multiscale and fine subspaces. In a first method, the multiscale basis functions are obtained by mapping coarse basis functions, based on corners used on primal iterative substructuring methods, to functions of global minimal energy. This approach delivers quasi-optimal a priori error energy approximation with respect to the mesh size, but it is not robust with respect to high-contrast coefficients. In a second method, edge modes based on local generalized eigenvalue problems are added to the corner modes. As a result, optimal a priori error energy estimate is achieved which is mesh and contrast independent. The methods converge at optimal rate even if the solution has minimum regularity, belonging only to the Sobolev space H¹. [ABSTRACT FROM AUTHOR]
Copyright of SIAM Journal on Numerical Analysis is the property of Society for Industrial & Applied Mathematics 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.)
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  Data: We consider finite element methods of multiscale type to approximate solutions for two-dimensional symmetric elliptic partial differential equations with heterogeneous L∞ coefficients. The methods are of Galerkin type and follow the Variational Multiscale and Localized Orthogonal Decomposition (LOD) approaches in the sense that it decouples spaces into multiscale and fine subspaces. In a first method, the multiscale basis functions are obtained by mapping coarse basis functions, based on corners used on primal iterative substructuring methods, to functions of global minimal energy. This approach delivers quasi-optimal a priori error energy approximation with respect to the mesh size, but it is not robust with respect to high-contrast coefficients. In a second method, edge modes based on local generalized eigenvalue problems are added to the corner modes. As a result, optimal a priori error energy estimate is achieved which is mesh and contrast independent. The methods converge at optimal rate even if the solution has minimum regularity, belonging only to the Sobolev space H¹. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
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  Group: Ab
  Data: <i>Copyright of SIAM Journal on Numerical Analysis is the property of Society for Industrial & Applied Mathematics 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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        Value: 10.1137/24M1665362
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      – Code: eng
        Text: English
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        PageCount: 23
        StartPage: 1055
    Subjects:
      – SubjectFull: Finite element method
        Type: general
      – SubjectFull: Elliptic differential equations
        Type: general
      – SubjectFull: Approximation error
        Type: general
      – SubjectFull: Multiple scale method
        Type: general
      – SubjectFull: Reduced-order models
        Type: general
      – SubjectFull: Galerkin methods
        Type: general
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      – TitleFull: SPECTRAL ACMS: A ROBUST LOCALIZED APPROXIMATED COMPONENT MODE SYNTHESIS METHOD.
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            – D: 01
              M: 05
              Text: 2025
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              Y: 2025
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