Adaptively-refinable polar-spline discrete differential forms: hierarchical construction, exactness, and applications

Abstract

A common approach to representing disk-like and sphere-like geometries in isogeometric analysis is to use collapsed-edge singularities in the underlying parameterization. The resulting tensor-product B-spline spaces on such singular parameterizations lack the required smoothness to be used in isogeometric analysis. Polar splines (Toshniwal et al., CMAME 2017) rectify this issue by extracting a smoother subspace, and have been successfully used to discretize high-order partial differential equations, as well as to perform structure-preserving discretizations of the de Rham complex.The latter is particularly relevant for mixed formulations that appear in applications such as electromagnetism and fluid dynamics. The main contributions of this paper are twofold. We combine the polar spline construction with hierarchical splines to obtain adaptively-refinable polar splines. Moreover, we do so in the context of structure-preserving methods. That is, we show how to construct hierarchical polar spline $k$-form spaces, prove that the corresponding basis functions are linearly independent, and prove that they form a cohomologically-correct discrete de Rham complex. These contributions provide a mathematically sound foundation for adaptive structure-preserving simulations on polar geometries. We also provide numerical experiments that validate the theory and illustrate the practical behavior of the construction with problems where preserving the structure is mandatory. These include a study of spurious harmonics that highlights how cohomology-breaking refinements differ between the polar and tensor-product settings, a discretization of a Maxwell eigenvalue problem on a locally-refined polar mesh, and an adaptive convergence study for a Hodge--Laplace problem.