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Quadrature rules for qualocation
Author(s) -
Junges Michael,
Schneider Claus
Publication year - 2003
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200310224
Subject(s) - quadrature (astronomy) , mathematics , gaussian quadrature , numerical integration , chebyshev filter , mathematical analysis , gauss–kronrod quadrature formula , chebyshev polynomials , boundary value problem , combinatorics , nyström method , physics , optics
Qualocation is a method for the numerical treatment of boundary integral equations on smooth curves which was developed by Chandler, Sloan and Wendland (1988‐2000) [1,2]. They showed that the method needs symmetric J–point–quadrature rules on [0, 1] that are exact for a maximum number of 1–periodic functions $$ G _{\alpha} (x) \ggleich \sum ^{\infty} _{k=1} k ^{-\alpha} \cos (2 \pi kx), \qquad \alpha > {1 \over {2}}. $$ The existence of 2–point–rules of that type was proven by Chandler and Sloan. For J ∈ {3, 4} such formulas have been calculated numerically in [2]. We show that the functions G α form a Chebyshev–system on [0, 1/2] for arbitrary indices á and thus prove the existence of such quadrature rules for any J.