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Multilevel methods for nonuniformly elliptic operators and fractional diffusion


Authors: Long Chen, Ricardo H. Nochetto, Enrique Otárola and Abner J. Salgado
Journal: Math. Comp. 85 (2016), 2583-2607
MSC (2010): Primary 65N55, 65F10, 65N22, 65N30, 35S15, 65N12
DOI: https://doi.org/10.1090/mcom/3089
Published electronically: March 3, 2016
MathSciNet review: 3522963
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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
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
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
Email: asalgad1@utk.edu

DOI: https://doi.org/10.1090/mcom/3089
Keywords: Finite elements, weighted Sobolev spaces, Muckenhoupt weights, anisotropic estimates, multilevel methods
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
Article copyright: © Copyright 2016 American Mathematical Society

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