论文标题
带有花纹的高阶几何方法：离散杂货店的分析 - 标准运算符
High order geometric methods with splines: an analysis of discrete Hodge--star operators
论文作者
论文摘要
提出了一种新型的样条几何方法方法。它的主要成分是使用良好的样条空间，形成一个离散的de rham综合体来构建一个原始序列$ \ {x^k_h \}^n_ {k = 0} $，从度数$ $ p $的键开始，以及双序列$ \ \ \ \ \ {x}^k_h \^$ jh $} n $ $ P-1 $。通过将均匀的边界条件施加到原始序列的空间，可以将两个序列相互映射成彼此。在此设置中，可以通过明确构建适当的离散版本的构成关系（称为hodge- star运营商）来最终适应许多熟悉的二阶部分偏微分方程。将提出基于样条空间之间全球和本地投影操作员的几种替代方案。该方法相对于类似发布的方法的吸引力是双重的：首先，它表现出高级收敛。其次，它不依赖于任何（拓扑）双网的几何实现。将采用各种空间维度中的几个数值示例来验证所提出的方法的中心思想，并将其特征与标准Galerkin方法进行等级分析中的特征进行比较。
A new kind of spline geometric method approach is presented. Its main ingredient is the use of well established spline spaces forming a discrete de Rham complex to construct a primal sequence $\{X^k_h\}^n_{k=0}$, starting from splines of degree $p$, and a dual sequence $\{\tilde{X}^k_h\}_{k=0}^n$, starting from splines of degree $p-1$. By imposing homogeneous boundary conditions to the spaces of the primal sequence, the two sequences can be isomorphically mapped into one another. Within this setup, many familiar second order partial differential equations can be finally accommodated by explicitly constructing appropriate discrete versions of constitutive relations, called Hodge--star operators. Several alternatives based on both global and local projection operators between spline spaces will be proposed. The appeal of the approach with respect to similar published methods is twofold: firstly, it exhibits high order convergence. Secondly, it does not rely on the geometric realization of any (topologically) dual mesh. Several numerical examples in various space dimensions will be employed to validate the central ideas of the proposed approach and compare its features with the standard Galerkin approach in Isogeometric Analysis.
