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The intersection of bivariate orthogonal polynomials on triangle patches


Koornwinder, Tom H; Sauter, Stefan A (2015). The intersection of bivariate orthogonal polynomials on triangle patches. Mathematics of Computation, 84(294):1795-1812.

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 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.

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 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 indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Date:July 2015
Deposited On:14 Jan 2016 12:16
Last Modified:05 Apr 2016 19:18
Publisher:American Mathematical Society
ISSN:0025-5718
Publisher DOI:https://doi.org/10.1090/S0025-5718-2014-02910-4

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Language: English
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