Romberg Type Cubature over Arbitrary Triangles

Walz, Guido

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URN: urn:nbn:de:bsz:180-madoc-17099
Document Type: Working paper
Year of publication: 1997
Publication language: English
Institution: School of Business Informatics and Mathematics > Sonstige - Fakultät für Mathematik und Informatik
MADOC publication series: Veröffentlichungen der Fakultät für Mathematik und Informatik > Institut für Mathematik > Mannheimer Manuskripte
Subject: 510 Mathematics
Classification: MSC: 65D30 41A60 65B05 ,
Subject headings (SWD): Kubatur , Romberg-Versuch , Numerische Integration , Triangulation , Extrapolation
Keywords (English): Romberg method, cubature , numerical integration , triangulation , polygon region , asymptotic expansion , extrapolation
Abstract: We develop an extrapolation algorithm for numerical integration over arbitary non-standard triangles in IR², which parallels the well-known univariate Romberg method. This is done by a suitable generalization of the trapezoidal rule over triangles, for which we can prove the existence of an asymptotic expansion. Our approach relies mainly on two ideas: The use of barycentric coordinates and the interpretation of the trapezoidal rule as the integral over an interpolating linear spline function. Since our method works for arbitrary triangles, it yields - via triangulation - a method for cubature over arbitrary, possibly non-convex, polygon regions in IR². Moreover, also numerical integration over convex polyhedra in IR d, d > 2 , can be accomplished without difficulties. Numerical examples show the stability and efficiency of the algorithm.
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