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 $A_2$--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.

References in corpus (3)

• Finite difference method for a general fractional porous medium equation
• A PDE approach to fractional diffusion in general domains: a priori error analysis
• Piecewise polynomial interpolation in Muckenhoupt weighted Sobolev spaces and applications

Cited by in corpus (2)

• A PDE approach to fractional diffusion: a posteriori error analysis
• Piecewise polynomial interpolation in Muckenhoupt weighted Sobolev spaces and applications