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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Multilevel methods for nonuniformly elliptic operators and fractional diffusion
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by Long Chen, Ricardo H. Nochetto, Enrique Otárola and Abner J. Salgado PDF
Math. Comp. 85 (2016), 2583-2607 Request permission

Abstract:

We develop and analyze multilevel methods for nonuniformly elliptic operators whose ellipticity holds in a weighted Sobolev space with an $A_2$–Muckenhoupt weight. Using the so-called Xu-Zikatanov (XZ) identity, we derive a nearly uniform convergence result under the assumption that the underlying mesh is quasi-uniform. As an application we also consider the so-called $\alpha$-harmonic extension to localize fractional powers of elliptic operators. Motivated by the scheme proposed by the second, third and fourth authors, we present a multilevel method with line smoothers and obtain a nearly uniform convergence result on anisotropic meshes. Numerical experiments illustrate the performance of our method.
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Additional Information
  • Long Chen
  • Affiliation: Department of Mathematics, University of California at Irvine, Irvine, California 92697
  • MR Author ID: 735779
  • Email: chenlong@math.uci.edu
  • Ricardo H. Nochetto
  • Affiliation: Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742
  • MR Author ID: 131850
  • Email: rhn@math.umd.edu
  • Enrique Otárola
  • Affiliation: Departamento de Matemática, Universidad Técnica Federico Santa María, Valparaíso, Chile
  • Email: enrique.otarola@usm.cl
  • Abner J. Salgado
  • Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996
  • MR Author ID: 847180
  • Email: asalgad1@utk.edu
  • Received by editor(s): March 17, 2014
  • Received by editor(s) in revised form: April 25, 2015
  • Published electronically: March 3, 2016
  • Additional Notes: The first author has been supported by NSF grants DMS-1115961, DMS-1418934, and DOE prime award # DE-SC0006903.
    The second and fourth authors have been supported in part by NSF grants DMS-1109325 and DMS-1411808.
    The third author was supported in part by the NSF grants DMS-1109325 and DMS-1411808 and by CONICYT through a CONICYT-FULBRIGHT Fellowship
    The fourth author was supported by NSF grant DMS-1418784
  • © Copyright 2016 American Mathematical Society
  • Journal: Math. Comp. 85 (2016), 2583-2607
  • MSC (2010): Primary 65N55, 65F10, 65N22, 65N30, 35S15, 65N12
  • DOI: https://doi.org/10.1090/mcom/3089
  • MathSciNet review: 3522963