Published

Multilevel methods for nonuniformly elliptic operators and fractional diffusion

L Chen, RH Nochetto, E Otarola, AJ Salgado

Mathematics of Computation, 2016

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ABSTRACT:

 We develop and analyze multilevel methods for nonuniformly
el- liptic operators whose ellipticity holds in a weighted Sobolev
space with an A2-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.