## The intersection of bivariate orthogonal polynomials on triangle patches

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- by Tom H. Koornwinder and Stefan A. Sauter PDF
- Math. Comp.
**84**(2015), 1795-1812 Request permission

## 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.## References

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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
- Email: T.H.Koornwinder@uva.nl
**Stefan A. Sauter**- Affiliation: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
- MR Author ID: 313335
- Email: stas@math.uzh.ch
- Received by editor(s): September 23, 2013
- Received by editor(s) in revised form: November 1, 2013
- Published electronically: December 18, 2014
- © Copyright 2014 American Mathematical Society
- Journal: Math. Comp.
**84**(2015), 1795-1812 - MSC (2010): Primary 65N50, 33C50; Secondary 65N15, 65N30, 33C45
- DOI: https://doi.org/10.1090/S0025-5718-2014-02910-4
- MathSciNet review: 3335892