Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz hp-DGFEM
Open access
Date
2012-11Type
- Report
ETH Bibliography
yes
Altmetrics
Abstract
We study the approximation of harmonic functions by means of harmonic polynomials in twodimensional, 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 $O(exp(-b\sqrt{N}))$, $N$ 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 $O(exp(-b \sqrt[3]{N}))$, and is due to the use of harmonic polynomial spaces, as opposed to complete polynomial spaces. Show more
Permanent link
https://doi.org/10.3929/ethz-a-010392193Publication status
publishedExternal links
Journal / series
SAM Research ReportVolume
Publisher
Seminar for Applied Mathematics, ETH ZurichOrganisational unit
03435 - Schwab, Christoph / Schwab, Christoph
03632 - Hiptmair, Ralf / Hiptmair, Ralf
Funding
247277 - Automated Urban Parking and Driving (EC)
Related publications and datasets
Is previous version of: http://hdl.handle.net/20.500.11850/78110
More
Show all metadata
ETH Bibliography
yes
Altmetrics