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.

Classification: 31A05, 30E10, 65N30

Keywords: approximation by harmonic polynomials, exponential orders of convergence, hp-finite elements

@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 - Mod\'elisation Math\'ematique et Analyse Num\'erique}, publisher = {EDP-Sciences}, volume = {48}, number = {3}, year = {2014}, pages = {727-752}, doi = {10.1051/m2an/2013137}, zbl = {1295.31004}, language = {en}, url = {http://www.numdam.org/item/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 - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 3, pp. 727-752. doi : 10.1051/m2an/2013137. http://www.numdam.org/item/M2AN_2014__48_3_727_0/

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