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Algebraic analysis of multigrid algorithms
Author(s) -
Pflaum Christoph
Publication year - 1999
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/(sici)1099-1506(199912)6:8<701::aid-nla181>3.0.co;2-l
Subject(s) - mathematics , rate of convergence , multigrid method , cauchy distribution , algebraic number , constant (computer programming) , convergence (economics) , additive schwarz method , bilinear interpolation , space (punctuation) , pure mathematics , mathematical analysis , partial differential equation , computer science , domain decomposition methods , finite element method , computer network , channel (broadcasting) , statistics , physics , economics , thermodynamics , programming language , economic growth , operating system
We study the convergence rate of multilevel algorithms from an algebraic point of view. This requires a detailed analysis of the constant in the strengthened Cauchy–Schwarz inequality between the coarse‐grid space and a so‐called complementary space. This complementary space may be spanned by standard hierarchical basis functions, prewavelets or generalized prewavelets. Using generalized prewavelets, we are able to derive a constant in the strengthened Cauchy–Schwarz inequality which is less than 0.31 for the L 2 and H 1 bilinear form. This implies a convergence rate less than 0.15. So, we are able to prove fast multilevel convergence. Furthermore, we obtain robust estimations of the convergence rate for a large class of anisotropic ellipic equations, even for some that are not H 1 ‐elliptic. Copyright © 1999 John Wiley & Sons, Ltd.