TY - JOUR
T1 - Algebraic Theory of Two-Grid Methods
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 2
SP - 168
EP - 198
PY - 2015
DA - 2015/08
SN - 8
DO - http://doi.org/10.4208/nmtma.2015.w04si
UR - https://global-sci.org/intro/article_detail/nmtma/12406.html
KW - 
AB - <p style="text-align: justify;">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.</p>