Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz hp-dGFEMHiptmair, 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
It is advisable to refer to the publisher's version if you intend to cite from this work. See Guidance on citing. To link to this item DOI: 10.1051/m2an/2013137 Abstract/SummaryWe 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.
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