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

Ralf Hiptmair; Andrea Moiola; Ilaria Perugia; Christoph Schwab

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2014)

  • Volume: 48, Issue: 3, page 727-752
  • ISSN: 0764-583X

Abstract

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We study the approximation of harmonic functions by means of harmonic polynomials in two-dimensional, bounded, star-shaped domains. Assuming that the functions possess analytic extensions to a δ-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 δ. We apply the obtained estimates to show exponential convergence with rate O(exp(-b√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(-b3√N)), and is due to the use of harmonic polynomial spaces, as opposed to complete polynomial spaces.

How to cite

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Hiptmair, Ralf, et al. "Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz hp-dGFEM." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.3 (2014): 727-752. <http://eudml.org/doc/273280>.

@article{Hiptmair2014,
abstract = {We study the approximation of harmonic functions by means of harmonic polynomials in two-dimensional, bounded, star-shaped domains. Assuming that the functions possess analytic extensions to a δ-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 δ. We apply the obtained estimates to show exponential convergence with rate O(exp(-b√N)), N being the number of degrees of freedom and b &gt; 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(-b3√N)), and is due to the use of harmonic polynomial spaces, as opposed to complete polynomial spaces.},
author = {Hiptmair, Ralf, Moiola, Andrea, Perugia, Ilaria, Schwab, Christoph},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {approximation by harmonic polynomials; exponential orders of convergence; hp-finite elements; exponential order of convergence; -finite elements},
language = {eng},
number = {3},
pages = {727-752},
publisher = {EDP-Sciences},
title = {Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz hp-dGFEM},
url = {http://eudml.org/doc/273280},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Hiptmair, Ralf
AU - Moiola, Andrea
AU - Perugia, Ilaria
AU - Schwab, Christoph
TI - Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz hp-dGFEM
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 3
SP - 727
EP - 752
AB - We study the approximation of harmonic functions by means of harmonic polynomials in two-dimensional, bounded, star-shaped domains. Assuming that the functions possess analytic extensions to a δ-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 δ. We apply the obtained estimates to show exponential convergence with rate O(exp(-b√N)), N being the number of degrees of freedom and b &gt; 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(-b3√N)), and is due to the use of harmonic polynomial spaces, as opposed to complete polynomial spaces.
LA - eng
KW - approximation by harmonic polynomials; exponential orders of convergence; hp-finite elements; exponential order of convergence; -finite elements
UR - http://eudml.org/doc/273280
ER -

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