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A Construction of $C^{r}$ Conforming Finite Elements on the Alfeld Split in Any Dimension

Ting Lin, Hendrik Speleers, Qingyu Wu

Abstract

Constructing $C^r$ conforming finite element spaces in any dimension is a long-standing problem. For general triangulations, this problem was recently addressed by Hu-Lin-Wu (2024), under certain conditions on supersmoothness and polynomial degree. In this paper, a first unified construction on the Alfeld split in any dimension is given, where the supersmoothness conditions and the polynomial degree requirement are relaxed.

A Construction of $C^{r}$ Conforming Finite Elements on the Alfeld Split in Any Dimension

Abstract

Constructing conforming finite element spaces in any dimension is a long-standing problem. For general triangulations, this problem was recently addressed by Hu-Lin-Wu (2024), under certain conditions on supersmoothness and polynomial degree. In this paper, a first unified construction on the Alfeld split in any dimension is given, where the supersmoothness conditions and the polynomial degree requirement are relaxed.

Paper Structure

This paper contains 9 sections, 25 theorems, 106 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Barycentric Coordinates and Multi-Index Sets
  3. Main Construction
  4. Pair of Refined Intrinsic Decompositions
  5. Properties of $Q$-Operators
  6. Proof of Unisolvence
  7. Proof of Continuity
  8. Dimension of Superspline Spaces
  9. Conclusion

Key Result

Theorem 1.1

Suppose that $\bm{r} := (r_{1}, \ldots, r_{d})$, $\rho$, $k$, and $b$ jointly satisfy asm:Cr-macro-element. Then, a $C^{r_1}$ conforming finite element space can be constructed with the local shape function space $\mathcal{S}_k^{\bm{r},\rho}(\mathcal{T}_A(K))$ defined in eq:shape-function. $\blacktr

Figures (5)

  • Figure 1: Alfeld split in two dimensions (Clough--Tocher split) and three dimensions.
  • Figure 2: Illustration of the notation for the vertices in the Alfeld split in two dimensions.
  • Figure 3: Illustration of the refined intrinsic decomposition of $\Sigma(\mathbb{I}_{d}, \bar{k})$ in \ref{['eq:decompose-full']} for $d=2$, $\bar{\bm{r}} = (1, 2)$, and $\bar{k} = 7$. The multi-indices related to a vertex are visualized as red disks, the multi-indices related to an edge as blue disks, and the remaining multi-indices as gray disks. Different shades of a color indicate different layers of multi-indices.
  • Figure 4: Illustration of the decomposition of $\Sigma^{\partial}$ in \ref{['eq:alpha-set']} (left) and of $\Sigma^{\circ}$ in \ref{['eq:beta-set']} (right) for $d = 2$, $\bm{r} = (3, 4)$, $\rho = 7$, $k = 9$, and $b = 2$. The multi-indices related to a vertex are visualized as red disks, the multi-indices related to an edge as blue disks, and the remaining multi-indices as gray disks. Different shades of a color indicate different layers of multi-indices.
  • Figure 5: Illustration of the decomposition of $\Sigma^{\partial}$ in \ref{['eq:alpha-set']} (left) and of $\Sigma^{\circ}$ in \ref{['eq:beta-set']} (right) for $d = 3$, $\bm{r} = (3, 4, 8)$, $\rho = 15$, $k = 17$, and $b = 2$. The multi-indices related to a vertex are visualized as red balls, the multi-indices related to an edge as blue balls, the multi-indices related to a facet as green balls, and the remaining multi-indices as gray balls. Different shades of a color indicate different layers of multi-indices.

Theorems & Definitions (33)

  • Theorem 1.1
  • Remark 2.1
  • Lemma 2.2
  • Definition 3.1: $Q$-Operator
  • Definition 3.2: Local degrees of freedom
  • Theorem 3.3
  • Theorem 3.4
  • Example 3.5: Dimension in the two-dimensional case
  • Example 3.6: Dimension in the three-dimensional case
  • Proposition 4.1: Refined intrinsic decomposition corresponding to $\bar{\bm{r}}$ hu2024construction
  • ...and 23 more