Volume 8, Issue 2
Algebraic Theory of Two-Grid Methods

Yvan Notay

Numer. Math. Theor. Meth. Appl.,8 (2015), pp. 168-198

Published online: 2015-08

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  • Abstract

About thirty years ago, Achi Brandt wrote a seminal paper providing a convergence theory for algebraic multigrid methods [Appl. Math. Comput., 19 (1986), pp. 23–56]. Since then, this theory has been improved and extended in a number of ways, and these results have been used in many works to analyze algebraic multigrid methods and guide their developments. This paper makes a concise exposition of the state of the art. Results for symmetric and nonsymmetric matrices are presented in a unified way, highlighting the influence of the smoothing scheme on the convergence estimates. Attention is also paid to sharp eigenvalue bounds for the case where one uses a single smoothing step, allowing straightforward application to deflation-based preconditioners and two-level domain decomposition methods. Some new results are introduced whenever needed to complete the picture, and the material is self-contained thanks to a collection of new proofs, often shorter than the original ones.

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@Article{NMTMA-8-168, author = {Yvan Notay}, title = {Algebraic Theory of Two-Grid Methods}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2015}, volume = {8}, number = {2}, pages = {168--198}, abstract = {

About thirty years ago, Achi Brandt wrote a seminal paper providing a convergence theory for algebraic multigrid methods [Appl. Math. Comput., 19 (1986), pp. 23–56]. Since then, this theory has been improved and extended in a number of ways, and these results have been used in many works to analyze algebraic multigrid methods and guide their developments. This paper makes a concise exposition of the state of the art. Results for symmetric and nonsymmetric matrices are presented in a unified way, highlighting the influence of the smoothing scheme on the convergence estimates. Attention is also paid to sharp eigenvalue bounds for the case where one uses a single smoothing step, allowing straightforward application to deflation-based preconditioners and two-level domain decomposition methods. Some new results are introduced whenever needed to complete the picture, and the material is self-contained thanks to a collection of new proofs, often shorter than the original ones.}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2015.w04si}, url = {http://global-sci.org/intro/article_detail/nmtma/12406.html} }

TY - JOUR T1 - Algebraic Theory of Two-Grid Methods AU - Yvan Notay JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 168 EP - 198 PY - 2015 DA - 2015/08 SN - 8 DO - http://dor.org/10.4208/nmtma.2015.w04si UR - https://global-sci.org/intro/article_detail/nmtma/12406.html KW - AB -

About thirty years ago, Achi Brandt wrote a seminal paper providing a convergence theory for algebraic multigrid methods [Appl. Math. Comput., 19 (1986), pp. 23–56]. Since then, this theory has been improved and extended in a number of ways, and these results have been used in many works to analyze algebraic multigrid methods and guide their developments. This paper makes a concise exposition of the state of the art. Results for symmetric and nonsymmetric matrices are presented in a unified way, highlighting the influence of the smoothing scheme on the convergence estimates. Attention is also paid to sharp eigenvalue bounds for the case where one uses a single smoothing step, allowing straightforward application to deflation-based preconditioners and two-level domain decomposition methods. Some new results are introduced whenever needed to complete the picture, and the material is self-contained thanks to a collection of new proofs, often shorter than the original ones.

Yvan Notay. (1970). Algebraic Theory of Two-Grid Methods. Numerical Mathematics: Theory, Methods and Applications. 8 (2). 168-198. doi:10.4208/nmtma.2015.w04si
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