Multilevel methods for nonuniformly elliptic operators
Abstract
We develop and analyze multilevel methods for nonuniformly elliptic operators whose ellipticity holds in a weighted Sobolev space with an --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. We also consider the so-called -harmonic extension to localize fractional powers of elliptic operators. Motivated by the scheme proposed in [R.H. Nochetto, E. Otarola, and A.J. Salgado. A PDE approach to fractional diffusion in general domains: a priori error analysis. arXiv:1302.0698, 2013] we present a multilevel method with line smoothers and obtain a nearly uniform convergence result on anisotropic meshes. Numerical experiments reveal a competitive performance of our method.
Cite
@article{arxiv.1403.4278,
title = {Multilevel methods for nonuniformly elliptic operators},
author = {Long Chen and Ricardo H. Nochetto and Enrique Otarola and Abner J. Salgado},
journal= {arXiv preprint arXiv:1403.4278},
year = {2014}
}