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

¹ Seminar of Applied Mathematics, ETH Zürich, 8092 Zürich, Switzerland.
hiptmair@math.ethz.ch
² Department of Mathematics and Statistics, University of Reading, Whiteknights, RG6 6AX, UK.
a.moiola@reading.ac.uk
³ Faculty of Mathematics, University of Vienna, 1090 Wien, Austria.
ilaria.perugia@univie.ac.at
⁴ Seminar of Applied Mathematics, ETH Zürich, 8092 Zürich, Switzerland.
schwab@math.ethz.ch

Received: 5 December 2012
Revised: 29 October 2013

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 > 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.