The Unisolvence of Lagrange Interpolation with Symmetric Interpolation Space and Nodes in High Dimension

TL;DR

This paper investigates the conditions for unisolvence in high-dimensional symmetric Lagrange interpolation, using group theory to improve understanding and computational efficiency in finite element methods.

Contribution

It provides a theoretical framework linking symmetry of nodes to unisolvence, enhancing both understanding and practical node selection in high-dimensional interpolation.

Findings

Necessary condition for unisolvence: symmetry of nodes set depends on the interpolation space

Group action and representation theories are used to characterize unisolvence conditions

Results reduce computational overhead in selecting interpolation nodes

Abstract

High-dimensional Lagrange interpolation plays a pivotal role in finite element methods, where ensuring the unisolvence and symmetry of its interpolation space and nodes set is crucial. In this paper, we leverage group action and group representation theories to precisely delineate the conditions for unisolvence. We establish a necessary condition for unisolvence: the symmetry of the interpolation nodes set is determined by the given interpolation space. Our findings not only contribute to a deeper theoretical understanding but also promise practical benefits by reducing the computational overhead associated with identifying appropriate interpolation nodes.