Symmetrizing grids, radial basis functions, and Chebyshev and Zernike polynomials for the D4 symmetry group; Interpolation within a squircle, Part I

A domain is invariant under the eight-element D4 symmetry group if it is unchanged by reflection with respect to the x and y axes and also the diagonal line x=y. Previous treatments of group theory for spectral methods have generally demanded a semesterʼs worth of group theory. We show this is unnecessary by providing explicit recipes for creating grids, etc. We show how to decompose an arbitrary function into six symmetry-invariant components, and thereby split the interpolation problem into six independent subproblems. We also show how to make symmetry-invariant basis functions from products of Chebyshev polynomials, from Zernike polynomials and from radial basis functions (RBFs) of any species. These recipes are completely general, and apply to any domain that is invariant under the dihedral group D4. These concepts are illustrated by RBF pseudospectral solutions of the Poisson equation in a domain bounded by a squircle, the square-with-rounded corners defined by x2ν+y2ν−1=0 where here ν=2. We also apply Chebyshev polynomials to compute eigenmodes of the Helmholtz equation on the square and show each mode belongs to one and only one of the six D4 classes.