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Exponential convergence of the hp version of isogeometric analysis in 1D

by A. Buffa and G. Sangalli and Ch. Schwab

(Report number 2012-39)

Abstract
We establish exponential convergence of the hp-version of isogeometric analysis for second order elliptic problems in one spacial dimension. Specifically, we construct, for functions which are piecewise analytic with a finite number of algebraic singularities at a-priori known locations in the closure of the open domain $\Omega$ of interest, a sequence $(\Pi^\ell_\sigma)_{\ell \ge 0}$ of interpolation operators which achieve exponential convergence. We focus on localized splines of reduced regularity so that the interpolation operators $(\Pi^\ell_\sigma)_{\ell \ge 0}$ are Hermite type projectors onto spaces of piecewise polynomials of degree $p \sim \ell$ whose differentiability increases linearly with $p$. As a consequence, the degree of conformity grows with $N$, so that asymptotically, the interpoland functions belong to $C^k(\Omega)$ for any fixed, finite $k$. Extensions to two- and to three-dimensional problems by tensorization are possible.

Keywords:

@Techreport{BSS12_492,
  author = {A. Buffa and G. Sangalli and Ch. Schwab},
  title = {Exponential convergence of the hp version of isogeometric analysis in 1D},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2012-39},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2012/2012-39.pdf },
  year = {2012}
}

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