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arXiv · 1403.4278

Multilevel methods for nonuniformly elliptic operators

Long Chen·Ricardo H. Nochetto·Enrique Otarola·Abner J. Salgado
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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. We also consider the so-called $\alpha$-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.

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Long Chen, Ricardo H. Nochetto, Enrique Otarola, Abner J. Salgado. 2014-03-17. Multilevel methods for nonuniformly elliptic operators. https://arxiv.org/abs/1403.4278

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