Journal
MATHEMATICS OF COMPUTATION
Volume 85, Issue 302, Pages 2583-2607Publisher
AMER MATHEMATICAL SOC
DOI: 10.1090/mcom/3089
Keywords
Finite elements; weighted Sobolev spaces; Muckenhoupt weights; anisotropic estimates; multilevel methods
Categories
Funding
- NSF [DMS-1115961, DMS-1418934, DMS-1109325, DMS-1418784, DMS-1411808]
- DOE prime award [DE-SC0006903]
- CONICYT through a CONICYT-FULBRIGHT Fellowship
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1418784] Funding Source: National Science Foundation
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1418934, 1411808] Funding Source: National Science Foundation
- U.S. Department of Energy (DOE) [DE-SC0006903] Funding Source: U.S. Department of Energy (DOE)
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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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