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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(-b 3√N)), and is due to the use of harmonic polynomial spaces, as opposed to complete polynomial spaces.
@article{M2AN_2014__48_3_727_0,
author = {Hiptmair, Ralf and Moiola, Andrea and Perugia, Ilaria and Schwab, Christoph},
title = {Approximation by harmonic polynomials in star-shaped domains and exponential convergence of {Trefftz} $hp${-dGFEM}},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {727--752},
publisher = {EDP-Sciences},
volume = {48},
number = {3},
year = {2014},
doi = {10.1051/m2an/2013137},
zbl = {1295.31004},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2013137/}
}
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 PY - 2014 SP - 727 EP - 752 VL - 48 IS - 3 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2013137/ DO - 10.1051/m2an/2013137 LA - en ID - M2AN_2014__48_3_727_0 ER -
%0 Journal Article %A Hiptmair, Ralf %A Moiola, Andrea %A Perugia, Ilaria %A Schwab, Christoph %T Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz $hp$-dGFEM %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2014 %P 727-752 %V 48 %N 3 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2013137/ %R 10.1051/m2an/2013137 %G en %F M2AN_2014__48_3_727_0
Hiptmair, Ralf; Moiola, Andrea; Perugia, Ilaria; Schwab, Christoph. Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz $hp$-dGFEM. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 3, pp. 727-752. doi: 10.1051/m2an/2013137
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