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Approximation by harmonic polynomials in star-shaped domainsand exponential convergence of Trefftz hp-dGFEM∗

Published online by Cambridge University Press:  11 February 2014

Ralf Hiptmair ,
Andrea Moiola ,
Ilaria Perugia  and
Christoph Schwab
Ralf Hiptmair
Affiliation:
Seminar of Applied Mathematics, ETH Zürich, 8092 Zürich, Switzerland.. [email protected]
Andrea Moiola
Affiliation:
Department of Mathematics and Statistics, University of Reading, Whiteknights, RG6 6AX, UK.; [email protected]
Ilaria Perugia
Affiliation:
Faculty of Mathematics, University of Vienna, 1090 Wien, Austria.; [email protected]
Christoph Schwab
Affiliation:
Seminar of Applied Mathematics, ETH Zürich, 8092 Zürich, Switzerland.; [email protected]
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Abstract

We study the approximation of harmonic functions by means of harmonic polynomials intwo-dimensional, bounded, star-shaped domains. Assuming that the functions possessanalytic extensions to a δ-neighbourhood of the domain, we proveexponential convergence of the approximation error with respect to the degree of theapproximating harmonic polynomial. All the constants appearing in the bounds are explicitand depend only on the shape-regularity of the domain and on δ. We applythe obtained estimates to show exponential convergence with rate O(exp(-b√N)), N being the number of degrees offreedom and b > 0, of a hp-dGFEM discretisation ofthe Laplace equation based on piecewise harmonic polynomials. This result is animprovement over the classical rate O(exp(-b 3√N)), and is due to the use of harmonic polynomialspaces, as opposed to complete polynomial spaces.

Keywords

Approximation by harmonic polynomialsexponential orders of convergencehp-finite elements
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 3 , May 2014 , pp. 727 - 752
Copyright
© EDP Sciences, SMAI, 2014

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