有限元外微分的旋转基：顶点置换下的闭式基变换

中文摘要 AI 辅助

单纯形上多项式微分形式空间 $\mathcal{P}_r\Lambda^1$ 和 $\mathcal{P}_r^-\Lambda^1$ 的几何分解基是重心标量多项式与方向性或 Whitney 1-形式的乘积。这些基依赖于单纯形顶点的排序。在非结构化网格上，相邻单元不必对该排序达成一致，这使得在有限元软件中实现协调性变得非平凡。现有策略通过数值计算自由度变换矩阵，或通过面泡空间或预处理网格来施加全局顶点排序。我们考虑 $\mathcal{P}_r^-\Lambda^1$（修剪）空间的标准基，并为 $\mathcal{P}_r\Lambda^1$（完整）空间引入一个新基，适用于任意多项式阶 $r \geq 1$ 和维数 $D \geq 2$，并将它们的基多项式指定为形函数。我们证明，在对称群 $S_{D+1}$ 中任意顶点重标号 $\pi$ 引起的坐标变换下，形函数的拉回具有闭式组合表达式。对于大多数基函数，该拉回是重标号排序的单个基函数，至多相差一个符号。在一个明确表征的“过滤命中”集合上，该拉回是至多 $D$ 个（完整空间）或恰好两个（修剪空间）重标号基函数的带符号和。所有系数对于所有 $r$、$D$ 和 $\pi$ 都在 $\{-1, +1\}$ 中。逆变换通过计算 $\pi^{-1}$ 处的公式获得，因此无需对基变换矩阵进行数值求逆。在具有任意顶点排序的单纯形网格上的协调组装和求值简化为索引操作。我们的结果与 Berchenko-Kogan 和 Licht 的重标号不变基研究进行了比较，并在开源 Julia 实现的 this http URL 库中得到了验证。

英文摘要

The geometrically decomposed bases of the polynomial differential form spaces $\mathcal{P}_rΛ^1$ and $\mathcal{P}_r^-Λ^1$ on a simplex are products of a barycentric scalar polynomial and a directional or Whitney 1-form. These bases depend on an ordering of the simplex vertices. On unstructured meshes, neighbouring cells need not agree on this ordering, making it non-trivial to achieve conformity in finite element software. Existing strategies compute degree-of-freedom transformation matrices numerically, or impose a global vertex ordering through face-bubble spaces or by preprocessing the mesh. We consider the standard basis of the $\mathcal{P}_r^-Λ^1$ (trimmed) space and introduce a new basis for the $\mathcal{P}_rΛ^1$ (full) space, for arbitrary polynomial order $r \geq 1$ and dimension $D \geq 2$, and prescribe their basis polynomials as shape functions. We show that the pullback of shape functions under the change of coordinates induced by an arbitrary vertex relabelling $π$ in the symmetric group $S_{D+1}$ admits a closed-form, combinatorial expression. For most basis functions this pullback is a single basis function of the relabelled ordering, up to sign. On an explicitly characterised "filter-hit" set, this pullback is a signed sum of at most $D$ (full space) or exactly two (trimmed space) relabelled basis functions. All coefficients are in $\{-1, +1\}$ for all $r$, $D$ and $π$. The inverse transformation is obtained by computing the formulas at $π^{-1}$, so no numerical inversion of a change-of-basis matrix is needed. Conforming assembly and evaluation on simplicial meshes with arbitrary vertex orderings reduce to index manipulation. Our results are compared with the study of relabelling-invariant bases of Berchenko-Kogan and Licht, and verified in an open-source Julia implementation in the Gridap.jl library.

发表机构

• Monash University（莫纳什大学）

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