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Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr


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1609-9389
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Volume 12, Issue 4

Issues

Polynomial of Best Uniform Approximation to 1/x and Smoothing in Two-level Methods

Johannes Kraus
  • Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences Altenberger Str. 69, 4040 Linz, Austria.
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Panayot Vassilevski
  • Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, P.O. Box 808, L-560, Livermore, CA 94551, U.S.A.
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Ludmil Zikatanov

Abstract

We derive defect correction scheme for constructing the sequence of polynomials of best approximation in the uniform norm to 1/x on a finite interval with positive endpoints. As an application, we consider two-level methods for scalar elliptic partial differential equation (PDE), where the relaxation on the fine grid uses the aforementioned polynomial of best approximation. Based on a new smoothing property of this polynomial smoother that we prove, combined with a proper choice of the coarse space, we obtain as a corollary, that the convergence rate of the resulting two-level method is uniform with respect to the mesh parameters, coarsening ratio and PDE coefficient variation.

Keywords: polynomial smoothing; best uniform approximation; two-level methods; elliptic problems with large coefficient variation

About the article

Received: 2012-05-08

Revised: 2012-08-28

Accepted: 2012-08-29

Published in Print:


Citation Information: Computational Methods in Applied Mathematics Comput. Methods Appl. Math., Volume 12, Issue 4, Pages 448–468, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.2478/cmam-2012-0026.

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Citing Articles

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[1]
S.K. Tomar
Computers & Mathematics with Applications, 2013, Volume 66, Number 6, Page 1024
[2]
Xiaozhe Hu, Panayot S. Vassilevski, and Jinchao Xu
Numerical Linear Algebra with Applications, 2016, Volume 23, Number 4, Page 746
[3]
Johannes Kraus, Raytcho Lazarov, Maria Lymbery, Svetozar Margenov, and Ludmil Zikatanov
SIAM Journal on Scientific Computing, 2016, Volume 38, Number 2, Page A875
[4]
Paola F. Antonietti, Marco Verani, and Ludmil Zikatanov
Computers & Mathematics with Applications, 2015, Volume 70, Number 11, Page 2674
[5]
J. Kraus, M. Lymbery, and S. Margenov
Numerical Linear Algebra with Applications, 2015, Volume 22, Number 6, Page 965
[6]
K.P.S. Gahalaut, S.K. Tomar, and J.K. Kraus
Computer Methods in Applied Mechanics and Engineering, 2013, Volume 266, Page 40
[7]
Ionela Moale and Veronika Pillwein
LMS Journal of Computation and Mathematics, 2014, Volume 17, Number 01, Page 233

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