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Grid transfer operators for multigrid methods
Author(s) -
Donatelli Marco
Publication year - 2011
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201110362
Subject(s) - multigrid method , toeplitz matrix , mathematics , partial differential equation , constant coefficients , generalization , convergence (economics) , elliptic partial differential equation , algebra over a field , mathematical analysis , pure mathematics , economics , economic growth
Local Fourier analysis (LFA) is a classical tool for proving convergence theorems for multigrid methods (MGMs). Analogously, the symbols of the involved matrices are studied to prove convergence results for MGMs for Toeplitz matrices. We show that in the case of elliptic partial differential equations (PDEs) with constant coefficients, the two different approaches lead to an equivalent optimality condition. We argue that the analysis for Toeplitz matrices is an algebraic generalization of the LFA which allows to deal not only with differential problems but also, e.g., with integral problems. A class of grid transfer operators related to the B‐spline's refinement equation is discussed as well. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)