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Nonlinear Approximation Rates and Besov Regularity for Elliptic PDEs on Polyhedral Domains

by M. Hansen

(Report number 2012-41)

Abstract
We investigate the Besov regularity for solutions of elliptic PDEs. This is based on regularity results in Babuska-Kondratiev spaces. Following the argument of Dahlke and DeVore, we first prove an embedding of these spaces into the scale $B^r_{\tau,\tau}(D)$ of Besov spaces with $\frac{1}{\tau}=\frac{r}{d}+\frac{1}{p}$. This scale is known to be closely related to $n$-term approximation w.r.to wavelet systems, and also adaptive Finite element approximation. Ultimately this yields the rate $n^{-r/d}$ for $u\in{\mathcal K}^m_{p,a}(D)\cap H^s_p(D)$ for $r$<$r^\ast\leq m$. In order to improve this rate to $n^{-m/d}$ we leave the scale $B^r_{\tau,\tau}(D)$ and instead consider the spaces $B^m_{\tau,\infty}(D)$. We determine conditions under which the space ${\mathcal K}^m_{p,a}(D)\cap H^s_p(D)$ is embedded into some space $B^m_{\tau,\infty}(D)$ for some $\frac{m}{d}+\frac{1}{p}>\frac{1}{\tau}\geq\frac{1}{p}$, which in turn indeed yields the desired $n$-term rate. As an intermediate step we also prove an extension theorem for Kondratiev spaces.

Keywords:

@Techreport{H12_494,
  author = {M. Hansen},
  title = {Nonlinear Approximation Rates and Besov Regularity for Elliptic PDEs on Polyhedral Domains},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2012-41},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2012/2012-41.pdf },
  year = {2012}
}

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