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Mathematics of Computation

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The intersection of bivariate orthogonal polynomials on triangle patches

Authors: Tom H. Koornwinder and Stefan A. Sauter
Journal: Math. Comp. 84 (2015), 1795-1812
MSC (2010): Primary 65N50, 33C50; Secondary 65N15, 65N30, 33C45
Published electronically: December 18, 2014
MathSciNet review: 3335892
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Abstract: In this paper, the intersection of bivariate orthogonal polynomials on triangle patches will be investigated. The result is interesting on its own but also has important applications in the theory of a posteriori error estimation for finite element discretizations with $ p$-refinement, i.e., if the local polynomial degree of the test and trial functions is increased to improve the accuracy. A triangle patch is a set of disjoint open triangles whose closed union covers a neighborhood of the common triangle vertex. On each triangle we consider the space of orthogonal polynomials of degree $ n$ with respect to the weight function which is the product of the barycentric coordinates. We show that the intersection of these polynomial spaces is the null space. The analysis requires the derivation of subtle representations of orthogonal polynomials on triangles. Up to four triangles have to be considered to identify that the intersection is trivial.

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Additional Information

Tom H. Koornwinder
Affiliation: Korteweg-de Vries Institute for Mathematics, University of Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, Netherlands

Stefan A. Sauter
Affiliation: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland

Received by editor(s): September 23, 2013
Received by editor(s) in revised form: November 1, 2013
Published electronically: December 18, 2014
Article copyright: © Copyright 2014 American Mathematical Society