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A Collocation Method for Integral Equations with Super-Algebraic Convergence Rate

Arens, Tilo; Rösch, Thomas

Abstract:

We consider biperiodic integral equations of the second kind with weakly singular kernels such as they arise in boundary integral equation methods. The equations are solved numerically using a collocation scheme based on trigonometric polynomials. The weak singularity is removed by a local change to polar coordinates. The resulting operators have smooth kernels and are discretized using the tensor product composite trapezodial rule. We prove stability and convergence of the scheme under suitable parameter choices, achieving algebraic convergence of any order under appropriate regularity assumptions. The method can be applied to typical boundary value problems such as potential and scattering problems both for bounded obstacles and for periodic surfaces. We present numerical results demonstrating that the expected convergence rates can be observed in practice.


Volltext §
DOI: 10.5445/IR/1000050980
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Angewandte und Numerische Mathematik (IANM)
Publikationstyp Forschungsbericht/Preprint
Publikationsjahr 2015
Sprache Englisch
Identifikator urn:nbn:de:swb:90-509802
KITopen-ID: 1000050980
Verlag Karlsruher Institut für Technologie (KIT)
Umfang 33 S.
Serie Preprint. Institut für Wissenschaftliches Rechnen und Mathematische Modellbildung, Karlsruher Institut für Technologie ; 2015,1
Schlagwörter numerical methods for integral equations, collocation method, super-algebraic convergence rate, Laplace equation, Helmholtz equation
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