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Comp."],"abstract":"<p>\n                    We develop and analyze multilevel methods for nonuniformly elliptic operators whose ellipticity holds in a weighted Sobolev space with an\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper A 2\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>A<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">A_2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    \u2013Muckenhoupt 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\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"alpha\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03b1\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\alpha<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -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.\n                  <\/p>","DOI":"10.1090\/mcom\/3089","type":"journal-article","created":{"date-parts":[[2015,12,30]],"date-time":"2015-12-30T09:46:26Z","timestamp":1451468786000},"page":"2583-2607","source":"Crossref","is-referenced-by-count":34,"title":["Multilevel methods for nonuniformly elliptic operators and fractional diffusion"],"prefix":"10.1090","volume":"85","author":[{"given":"Long","family":"Chen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ricardo","family":"Nochetto","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Enrique","family":"Ot\u00e1rola","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Abner","family":"Salgado","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2016,3,3]]},"reference":[{"issue":"5","key":"1","doi-asserted-by":"publisher","first-page":"1993","DOI":"10.1137\/S0036142900375414","article-title":"Multigrid methods for anisotropic edge refinement","volume":"40","author":"Apel, Thomas","year":"2002","journal-title":"SIAM J. 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