A comparative study of several classical, discrete differential and isogeometric methods for solving Poisson's equation on the disk

Abstract

This paper outlines and qualitatively compares the implementations of seven different methods for solving Poisson's equation on the disk. The methods include two classical finite elements, a cotan formula-based discrete differential geometry approach and four isogeometric constructions. The comparison reveals numerical convergence rates and, particularly for isogeometric constructions based on Catmull-Clark elements, the need to carefully choose quadrature formulas. The seven methods include two that are new to isogeometric analysis. Both new methods yield O(h3) convergence in the L2 norm, also when points are included where n ≠ 4 pieces meet. One construction is based on a polar, singular parameterization; the other is a G1 tensor-product construction.