TY - JOUR

T1 - Polynomial of best uniform approximation to 1/x and smoothing in two-level methods

AU - Kraus, Johannes

AU - Vassilevski, Panayot

AU - Zikatanov, Ludmil

N1 - Funding Information:
The work of the first author has been supported by the Austrian Science Fund, Grant P22989-N18. The work of the second author is performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. The work of the third author is supported in part by the National Science Foundation DMS-0810982, U.S. Department of Energy grant DE-SC0006903 and Lawrence Livermore National Laboratory subcontract B595949.

PY - 2012/10

Y1 - 2012/10

N2 - 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.

AB - 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.

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U2 - 10.2478/cmam-2012-0026

DO - 10.2478/cmam-2012-0026

M3 - Article

AN - SCOPUS:84868622600

VL - 12

SP - 448

EP - 468

JO - Computational Methods in Applied Mathematics

JF - Computational Methods in Applied Mathematics

SN - 1609-4840

IS - 4

ER -