Multilevel methods for nonuniformly elliptic operators and fractional diffusion

Chen, Long; Nochetto, Ricardo; Otárola, Enrique; Salgado, Abner

doi:10.1090/mcom/3089

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;

Math. Comp. 85 (2016), 2583-2607

DOI: https://doi.org/10.1090/mcom/3089

Published electronically: March 3, 2016

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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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