Polynomial Interpolation And Application Of Functions In A Squircle Domain

Single-domain spectral methods have been largely restricted to square domain, cir-cle domain, triangular domain and so on. Here we study a new domain:a squircle domain. The zero isoline of it is B(x,y)=x2v+y2v-1. The boundary varies smoothly from a circle (v=1) to the square (v=∞). It is worth studying the squircle domain. A domain is invariant under the eight-element D4symmetry group if it is unchanged by reflection with respect to the x and y axes and also the diagonal line x=y. Previous treat-ments of group theory for spectral methods have generally demanded a semester’s 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 subprob-lems. 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. 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 D4classes.Polynomial least-squares hyperinterpolation converges geometrically as long as the number of points P is (at least) double the number of basis functions N. The polyno-mial grid was made denser near the boundaries ("Chebyshev-like") by depositing grid points along wisely chosen contours of B. Gaussian radial basis functions (RBFs) were more robust in the sense that they, too, converged geometrically, but hyperinterpolation (P> N) and a Chebyshevized grid were unnecessary. A uniform grid, truncated to in-clude only those points within the squircle, was satisfactory even without interpolation points on the boundary (although boundary points are a cost-effective improvement). For a given number of points P, however, RBF interpolation was only slightly more accurate than polynomial hyperinterpolation, and needed twice as many basis functions. Interpo-lation costs can be greatly reduced by exploiting the invariance of the squircle-bounded domain to the eight elements D4dihedral group.