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Unisolvency for Polynomial Interpolation in Simplices with Symmetrical Nodal Distributions

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Abstract

In one dimension, nodal locations that are distinct are necessary and sufficient to ensure that a unique polynomial interpolant exists for data provided at a set of nodes, i.e. that the set of nodes is unisolvent. In multiple dimensions however, unisolvency for a polynomial interpolant of degree p is not ensured even with nodal locations that are distinct and a set of n nodes, with n equal to the cardinality of a set of polynomial basis functions of at most degree p. In this paper a set of equations is derived for simplices of one to three dimensions with symmetrical nodal distributions to identify a combination of symmetry orbits that can provide a unisolvent set of nodes. The results suggest that there is a unique combination of symmetry orbits that can provide a unisolvent set of nodes for each degree of polynomial interpolant. Consequently, all other combinations of symmetry orbits cannot provide a unisolvent set of nodes for a degree p polynomial interpolant. This is verified numerically up to degree 10 for triangles and degree 7 for tetrahedra. The results suggest that the same is also true for higher-order polynomial interpolants. This significantly reduces the number of combination of symmetry orbits that needs to be considered. For example, for a tetrahedron with a degree seven interpolant, only one combination of symmetry orbits needs to be considered instead of the 161 different combinations of symmetry orbits that provide a set of nodes with n equal to the cardinality of the set of basis functions of at most degree seven. For a symmetrical nodal distribution in a simplex, the conditions presented are necessary but not sufficient to have a unisolvent set of nodes for polynomial interpolation.

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Acknowledgements

The authors would like to thank Tristan Montoya, Prof. Masayuki Yano, and Prof. David Del Rey Fernández for several helpful discussions and for their help to improve this paper. Furthermore, the authors would like to thank the anonymous reviewer for several helpful suggestions, which helped significantly improve this paper. Finally, the authors acknowledge the financial support provided by the Government of Ontario through the Ontario Graduate Scholarship and the Government of Canada for the Vanier Canada Graduate Scholarship.

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  1. University of Toronto Institute for Aerospace Studies, Toronto, ON, Canada

    André L. Marchildon & David W. Zingg

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  1. André L. Marchildon
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Correspondence to André L. Marchildon.

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Marchildon, A.L., Zingg, D.W. Unisolvency for Polynomial Interpolation in Simplices with Symmetrical Nodal Distributions. J Sci Comput 92, 50 (2022). https://doi.org/10.1007/s10915-022-01904-w

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  • DOI: https://doi.org/10.1007/s10915-022-01904-w

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