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The Qualocation Method for Symm's Integral Equation on a Polygon
Author(s) -
Elschner J.,
Prössdorf S.,
Sloan I. H.
Publication year - 1996
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19961770107
Subject(s) - mathematics , superconvergence , discretization , polygon (computer graphics) , quadrature (astronomy) , mathematical analysis , integral equation , convergence (economics) , nonlinear system , geometry , finite element method , telecommunications , physics , frame (networking) , quantum mechanics , economic growth , computer science , electrical engineering , economics , thermodynamics , engineering
This paper discusses the convergence of the qualocation method for Symm's integral equation on closed polygonal boundaries in IR 2 . Qualocation is a Petrov‐Galerkin method in which the outer integrals are performed numerically by special quadrature rules. Before discretisation a nonlinear parametrisation of the polygon is introduced which varies more slowly than arc‐length near each corner and leads to a transformed integral equation with a regular solution. We prove that the qualocation method using smoothest splines of any order k on a uniform mesh (with respect to the new parameter) converges with optimal order O ( h k ). Furthermore, the method is shown to produce superconvergent approximations to linear functionals, retaining the same high convergence rates as in the case of a smooth curve.