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A new embedding result for Kondratiev spaces and application to adaptive approximation of elliptic PDEs

by M. Hansen

(Report number 2014-30)

Abstract
In a continuation of recent work on Besov regularity of solutions to elliptic PDEs in Lipschitz domains with polyhedral structure, we prove an embedding between weighted Sobolev spaces (Kondratiev spaces) relevant for the regularity theory for such elliptic problems, and Triebel-Lizorkin spaces, which are known to be closely related to approximation spaces for nonlinear \(n\)-term wavelet approximation. Additionally, we also provide necessary conditions for such embeddings. As a further application we discuss the relation of these embedding results with results by Gaspoz and Morin for approximation classes for adaptive Finite element approximation, and subsequently apply these result to parametric problems.

Keywords: Regularity for elliptic PDEs, Kondratiev spaces, Besov regularity, Triebel-Lizorkin spaces, wavelet decomposition, $n$-term approximation, adaptive Finite element approximation, parametric elliptic problems

@Techreport{H14_580,
  author = {M. Hansen},
  title = {A new embedding result for Kondratiev spaces and application to adaptive approximation of elliptic PDEs},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2014-30},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2014/2014-30.pdf },
  year = {2014}
}

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