The intersection of bivariate orthogonal polynomials on triangle patches

TL;DR

This paper investigates the intersection of bivariate orthogonal polynomials on triangle patches, revealing that the intersection is trivial, which has implications for finite element error estimation with p-refinement.

Contribution

It demonstrates that the intersection of polynomial spaces on triangle patches is null, providing new insights into orthogonal polynomial representations and their applications in finite element methods.

Findings

The intersection of polynomial spaces on triangle patches is trivial.

Orthogonal polynomials on triangles can be represented through subtle formulas.

Up to four triangles are needed to confirm the trivial intersection.

Abstract

In this paper, the intersection of bivariate orthogonal polynomials on triangle patches will be investigated. The result is interesting by 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.