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Multigrid Methods of Macrogrid Domain Decomposition.

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Title: Multigrid Methods of Macrogrid Domain Decomposition.
Authors: Il'in, V. P.1 (AUTHOR) ilin@sscc.ru
Source: Computational Mathematics & Mathematical Physics. Jun2025, Vol. 65 Issue 6, p1220-1231. 12p.
Subjects: Multigrid methods (Numerical analysis), Domain decomposition methods, Algebraic equations, Krylov subspace, Iterative methods (Mathematics), Boundary value problems, Sparse matrices
Abstract: Integrated domain decomposition multigrid methods (DDM-MG) for solving large systems of linear algebraic equations (SLAEs) with sparse symmetric or asymmetric matrices are considered. Such systems are obtained as a result of grid approximations of multidimensional boundary value problems. The proposed algorithms are based on the construction of single-layer or two-layer macrogrids and a special ordering of nodes according to their belonging to various topological primitives of the macrogrid: macronodes, macroedges, macrofaces, and subdomains. With a consistent numbering of vector components, the SLAE matrix in the three-dimensional case takes a block-tridiagonal form of the fourth order. To solve it, an iterative preconditioned method in Krylov subspaces is used. In this case, the solution of auxiliary systems in subdomains is carried out by multigrid methods of block incomplete factorization based on a similar topologically oriented ordering of nodes but at the microlevel rather than at the macrolevel, as a result of which a single preconditioner of a recursively nested type is formed. The justification of the proposed methods is carried out for matrices of the Stieltjes type. [ABSTRACT FROM AUTHOR]
Copyright of Computational Mathematics & Mathematical Physics 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.)
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  Data: Integrated domain decomposition multigrid methods (DDM-MG) for solving large systems of linear algebraic equations (SLAEs) with sparse symmetric or asymmetric matrices are considered. Such systems are obtained as a result of grid approximations of multidimensional boundary value problems. The proposed algorithms are based on the construction of single-layer or two-layer macrogrids and a special ordering of nodes according to their belonging to various topological primitives of the macrogrid: macronodes, macroedges, macrofaces, and subdomains. With a consistent numbering of vector components, the SLAE matrix in the three-dimensional case takes a block-tridiagonal form of the fourth order. To solve it, an iterative preconditioned method in Krylov subspaces is used. In this case, the solution of auxiliary systems in subdomains is carried out by multigrid methods of block incomplete factorization based on a similar topologically oriented ordering of nodes but at the microlevel rather than at the macrolevel, as a result of which a single preconditioner of a recursively nested type is formed. The justification of the proposed methods is carried out for matrices of the Stieltjes type. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Computational Mathematics & Mathematical Physics 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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        Value: 10.1134/S0965542525700460
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      – Code: eng
        Text: English
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        Type: general
      – SubjectFull: Domain decomposition methods
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      – SubjectFull: Algebraic equations
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      – SubjectFull: Krylov subspace
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      – SubjectFull: Iterative methods (Mathematics)
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      – SubjectFull: Boundary value problems
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      – SubjectFull: Sparse matrices
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      – TitleFull: Multigrid Methods of Macrogrid Domain Decomposition.
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              Text: Jun2025
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