$\mathbb{R}^n$中分层B样条离散de Rham复形精确性的局部可验证充分条件
Locally-verifiable sufficient conditions for exactness of the hierarchical B-spline discrete de Rham complex in $\mathbb{R}^n$
摘要：
给定一个域$Ω\subset \mathbb{R}^n$，在研究定义在$Ω$上的电磁学和流体力学问题时，微分形式的de Rham复形会自然出现，它的离散化有助于为这类问题建立稳定的数值方法。为了构造这种稳定的方法，一个关键的要求是确保离散子复形与连续复形上同调等价。当$Ω$是超立方体时，我们要求离散子复形是精确的。针对这类$Ω$，我们从理论上分析了由分级B样条微分形式构造的离散de Rham复形，即，离散微分形式是平滑样条，并且支持自适应细化这些特性是实现精确和有效的数值模拟的关键。我们提供了本地可验证的充分条件，确保离散样条复杂的是准确的。数值试验支持的理论结果，讨论的例子包括复杂的，满足我们规定的条件，以及那些违反这些条件。
英文摘要：
Given a domain $Ω\subset \mathbb{R}^n$, the de Rham complex of differential forms arises naturally in the study of problems in electromagnetism and fluid mechanics defined on $Ω$, and its discretization helps build stable numerical methods for such problems. For constructing such stable methods, one critical requirement is ensuring that the discrete subcomplex is cohomologically equivalent to the continuous complex. When $Ω$ is a hypercube, we thus require that the discrete subcomplex be exact. Focusing on such $Ω$, we theoretically analyze the discrete de Rham complex built from hierarchical B-spline differential forms, i.e., the discrete differential forms are smooth splines and support adaptive refinements - these properties are key to enabling accurate and efficient numerical simulations. We provide locally-verifiable sufficient conditions that ensure that the discrete spline complex is exact. Numerical tests are presented to support the theoretical results, and the examples discussed include complexes that satisfy our prescribed conditions as well as those that violate them.
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