发表机构
Delft Institute of Applied Mathematics, Delft University of Technology; Department of Applied Mathematics, University of Santiago de Compostela; Galician Centre for Mathematical Research and Technology (CITMAga), University of Santiago de Compostela（代尔夫特应用数学研究所，代尔夫特理工大学; 圣地亚哥德孔波斯特拉大学应用数学系; 圣地亚哥德孔波斯特拉大学加利西亚数学研究与技术研究中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文将极样条与分层样条结合得到可自适应细化的极样条，构造出分层极样条k-形式空间，证明其基函数线性无关且构成上同调正合的离散de Rham复形，通过数值实验验证理论并展示其在保结构模拟中的应用。
AI 中文摘要
在等几何分析中，表示盘状和球状几何的常用方法是在基础参数化中使用缩边奇点。这类奇异参数化上生成的张量积B样条空间缺乏等几何分析所需的光滑性，而极样条（Toshniwal等人，CMAME 2017）通过提取更光滑的子空间解决了该问题，已成功用于离散高阶偏微分方程，以及对de Rham复形进行保结构离散，后者对电磁学、流体动力学等应用中出现的混合公式尤为重要。本文的主要贡献有两点：一是将极样条构造与分层样条结合，得到可自适应细化的极样条，且该构造在保结构方法的框架下完成；二是展示如何构造分层极样条k-形式空间，证明对应的基函数线性无关，并证明它们构成上同调正合的离散de Rham复形。这些贡献为极几何上的自适应保结构模拟提供了数学上严谨的基础。本文还提供了数值实验，验证了该理论并说明构造的实际表现，涉及必须保结构的问题，包括虚假谐波研究（凸显极坐标与张量积设置中破坏上同调的细化方式的差异）、局部细化极网格上的Maxwell本征问题离散，以及Hodge-Laplace问题的自适应收敛研究。
英文摘要
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.
