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Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz hp-DGFEM

by R. Hiptmair and A. Moiola and I. Perugia and Ch. Schwab

(Report number 2012-38)

Abstract
We study the approximation of harmonic functions by means of harmonic polynomials in twodimensional, bounded, star-shaped domains. Assuming that the functions possess analytic extensions to a $\delta$-neighbourhood of the domain, we prove exponential convergence of the approximation error with respect to the degree of the approximating harmonic polynomial. All the constants appearing in the bounds are explicit and depend only on the shape-regularity of the domain and on $\delta$. We apply the obtained estimates to show exponential convergence with rate $O(exp(-b\sqrt{N}))$, $N$ being the number of degrees of freedom and $b > 0$, of a hp-dGFEM discretisation of the Laplace equation based on piecewise harmonic polynomials. This result is an improvement over the classical rate $O(exp(-b ^3\sqrt{N}))$, and is due to the use of harmonic polynomial spaces, as opposed to complete polynomial spaces.

Keywords:

@Techreport{HMPS12_487,
  author = {R. Hiptmair and A. Moiola and I. Perugia and Ch. Schwab},
  title = {Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz hp-DGFEM},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2012-38},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2012/2012-38.pdf },
  year = {2012}
}

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