Cubature rules for harmonic functions based on Radon projections
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We construct a class of cubature formulae for harmonic functions on the unit disk based on line integrals over \(2n+1\) distinct chords. These chords are assumed to have constant distance \(t\) to the center of the disk, and their angles to be equispaced over the interval \([0,2\pi ]\). If \(t\) is chosen properly, these formulae integrate exactly all harmonic polynomials of degree up to \(4n+1\), which is the highest achievable degree of precision for this class of cubature formulae. For more generally distributed chords, we introduce a class of interpolatory cubature formulae which we show to coincide with the previous formulae for the equispaced case. We give an error estimate for a particular cubature rule from this class and provide numerical examples.
KeywordsCubature rules Harmonic functions Radon projections
Mathematics Subject Classification (2000)65D32 65D05 41A55
The authors express their gratitude to the two anonymous referees whose remarks and corrections improved the quality of the paper significantly. The authors acknowledge the support by Bulgarian National Science Fund, Grant DNTS/Austria 01/6 and Grant DFNI-T01/0001. The second author was supported by the project AComIn “Advanced Computing for Innovation”, Grant 316087, funded by the FP7 Capacity Programme “Research Potential of Convergence Regions”.
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