Spectral methods in non-tensor geometry, Part II: Chebyshev versus Zernike polynomials, gridding strategies and spectral extension on squircle-bounded and perturbed-quadrifolium domains

Abstract

Single-domain spectral methods have been largely restricted to tensor product bases on a tensor product grid. To break the “tensor barrier”, we studied approximation in two idealized families of domains. One family is bounded by a “squircle”, the zero isoline of B(x,y)=x2ν+y2ν−1. The boundary varies smoothly from a circle [ν=1] to the square [ν=∞]. The other family is bounded by a “perturbed quadrifolium”, the plane algebraic curve δ(x2+y2)−((x2+y2)3−(x2−y2)2); this varies smoothly from the singular, self-intersecting curve known as the quadrifolium to a circle as δ varies from zero to infinity. We compared two different bivariate polynomial bases, truncating by total degree. Zernike polynomials are natural for the disk; tensor products of Chebyshev polynomials are equally sensible for the square. Both yield an exponential rate of convergence for our non-tensor, neither disk-nor-square domains; indeed, the Chebyshev basis worked well for the disk and the Zernike polynomials were good for the square. The expected differences due to numerical ill-conditioning did not emerge, much to our surprise. The price for the nontensor domain was that hyperinterpolation was necessary, that is, least squares fitting with more interpolation constraints than unknowns. Denoting the number of interpolation points by P and the basis size by N, a ratio of P/N around two to three was optimum while P near one was very inacccurate. A uniform grid, truncated to include only those points within the squircle or other boundary curve, was satisfactory even without interpolation points on the boundary (although boundary points are a cost-effective improvement). Interpolation costs were greatly reduced by exploiting the invariance of the squircle-bounded and perturbed-quadrifolium domains to the eight element D4 dihedral group.