Volume 48, Number 3, May-June 2014
|Page(s)||727 - 752|
|Published online||11 February 2014|
Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz hp-dGFEM∗
1 Seminar of Applied Mathematics, ETH Zürich, 8092 Zürich,
2 Department of Mathematics and Statistics, University of Reading, Whiteknights, RG6 6AX, UK.
3 Faculty of Mathematics, University of Vienna, 1090 Wien, Austria.
4 Seminar of Applied Mathematics, ETH Zürich, 8092 Zürich, Switzerland.
Revised: 29 October 2013
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.
Mathematics Subject Classification: 31A05 / 30E10 / 65N30
Key words: Approximation by harmonic polynomials / exponential orders of convergence / hp-finite elements
© EDP Sciences, SMAI, 2014
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.