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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]
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Database: Engineering Source
Description
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]
ISSN:00361429
DOI:10.1137/24M1665362