Cubature formula and interpolation on the cubic domain

TL;DR

This paper develops new cubature formulas and interpolation methods on cubic domains using discrete Fourier analysis, providing explicit formulas and node counts for efficient numerical integration and interpolation in 2D and 3D.

Contribution

It introduces novel derivations of Gaussian-type cubature formulas and explicit fundamental interpolation polynomials on cubic domains based on lattice tiling and Fourier analysis.

Findings

Derived new cubature formulas for $[-1,1]^2$ and $[-1,1]^3$

Provided explicit formulas for fundamental interpolation polynomials

Achieved node counts of approximately $n^3/4$ for 3D cubature

Abstract

Several cubature formulas on the cubic domains are derived using the discrete Fourier analysis associated with lattice tiling, as developed in \cite{LSX}. The main results consist of a new derivation of the Gaussian type cubature for the product Chebyshev weight functions and associated interpolation polynomials on $[-1,1]^2$, as well as new results on $[-1,1]^3$. In particular, compact formulas for the fundamental interpolation polynomials are derived, based on $n^3/4 +\CO(n^2)$ nodes of a cubature formula on $[-1,1]^3$.