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

Hiptmair, R., Moiola, A., Perugia, I. and Schwab, C. (2014) Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz hp-dGFEM. ESAIM: Mathematical Modelling and Numerical Analysis M2AN, 48 (3). pp. 727-752. ISSN 1290-3841 doi: 10.1051/m2an/2013137

Abstract/Summary

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 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 square root 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 cubic root N )), and is due to the use of harmonic polynomial spaces, as opposed to complete polynomial spaces.

Item Type	Article
URI	https://reading-clone.eprints-hosting.org/id/eprint/36000
Identification Number/DOI	10.1051/m2an/2013137
Refereed	Yes
Divisions	Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
Publisher	EDP Sciences
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