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.
UR - http://www.scopus.com/inward/record.url?scp=84868622600&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84868622600&partnerID=8YFLogxK
U2 - 10.2478/cmam-2012-0026
DO - 10.2478/cmam-2012-0026
M3 - Article
AN - SCOPUS:84868622600
SN - 1609-4840
VL - 12
SP - 448
EP - 468
JO - Computational Methods in Applied Mathematics
JF - Computational Methods in Applied Mathematics
IS - 4
ER -