论文标题
在$ \ mathbb {r}^n $中，本地可验证的足够条件，可用于精确性的精确性。
Locally-verifiable sufficient conditions for exactness of the hierarchical B-spline discrete de Rham complex in $\mathbb{R}^n$
论文作者
论文摘要
给定域$ω\ subset \ mathbb {r}^n $，差异形式的de rham复合物自然出现在$ω$上定义的电磁和流体力学问题的研究中，其离散化有助于构建稳定的数值方法，以解决此类问题。为了构建这种稳定的方法，一个关键的要求是确保离散子复合物在共同体学上等同于连续复合物。因此，当$ω$是超立方体时，我们要求确切的离散子复合。为了关注这样的$ω$，我们理论上分析了由层次B -Spline差异形式构建的离散DE RHAM复合物，即离散的差异形式是光滑的花键和支持自适应改进 - 这些属性是实现准确有效的数值模拟的关键。我们提供当地可验证的足够条件，以确保离散的样条络合物精确。提出了数值测试以支持理论结果，讨论的示例包括满足我们规定条件以及违反它们的条件的复合物。
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.
