Multilevel additive and multiplicative methods for orthogonal spline collocation problems |
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Authors: | Bernard Bialecki Maksymilian Dryja |
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Institution: | (1) Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, Colorado 80401, USA; e-mail: bbialeck@mines.edu , US;(2) Department of Mathematics, Warsaw University, Banacha 2, 02-097 Warsaw, Poland , PL |
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Abstract: | Summary. Multilevel preconditioners are proposed for the iterative solution of the discrete problems which arise when orthogonal spline
collocation with piecewise Hermite bicubics is applied to the Dirichlet boundary value problem for a self-adjoint elliptic
partial differential equation on a rectangle. Additive and multiplicative preconditioners are defined respectively as sums
and products of independent operators on a sequence of nested subspaces of the fine partition approximation space. A general
theory of additive and multiplicative Schwarz methods is used to prove that the preconditioners are spectrally equivalent
to the collocation discretization of the Laplacian with the spectral constants independent of the fine partition stepsize
and the number of levels. The preconditioned conjugate gradient and preconditioned Orthomin methods are considered for the
solution of collocation problems. An implementation of the methods is discussed and the results of numerical experiments are
presented.
Received March 1, 1994 / Revised version received January 23, 1996 |
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Keywords: | Mathematics Subject Classification (1991):65F10 65N30 |
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