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The sharpness condition for constructing a finite element from a superspline

Jun Hu, Ting Lin, Qingyu Wu, Beihui Yuan

Abstract

This paper addresses sharpness conditions for constructing $C^r$ conforming finite element spaces from a superspline spaces on general simplicial triangulations. We introduce the concept of extendability for the pre-element spaces, which encompasses both the superspline spaces and the finite element spaces. By examining the extendability condition for both types of spaces, we provide an answer to the conditions regarding the construction. A corollary of our results is that constructing $C^r$ conforming elements in $d$ dimensions generally requires an extra $C^{2^{s}r}$ continuity on $s$-codimensional simplices, and the polynomial degree is at least $(2^d r + 1)$.

The sharpness condition for constructing a finite element from a superspline

Abstract

This paper addresses sharpness conditions for constructing conforming finite element spaces from a superspline spaces on general simplicial triangulations. We introduce the concept of extendability for the pre-element spaces, which encompasses both the superspline spaces and the finite element spaces. By examining the extendability condition for both types of spaces, we provide an answer to the conditions regarding the construction. A corollary of our results is that constructing conforming elements in dimensions generally requires an extra continuity on -codimensional simplices, and the polynomial degree is at least .
Paper Structure (5 sections, 12 theorems, 58 equations, 4 figures)

This paper contains 5 sections, 12 theorems, 58 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Extendability: An example
  3. Proof of \ref{['thm:extendable-srk']}
  4. Macro superspline spaces in two dimensions
  5. Details of the finite element spaces and the coincidence with the superspline spaces

Key Result

Theorem 1.1

Let $\mathsf S_k^{\bm r} : \mathcal{T} \mapsto \bm S_k^{\bm r}(\mathcal{T})$ be a superspline space mapping. Then, $\mathsf S_k^{\bm r}$ admits a construction of a finite element if and only if the continuity vector $\bm{r}$ and the polynomial degree $k$ satisfy Assumption eq:assumption.

Figures (4)

  • Figure 1: A mesh containing a singular vertex.
  • Figure 2: A two-dimensional example supporting \ref{['prop:k-rd']}.
  • Figure 3: Three-dimensional illustrations of $\mathcal{T}'$ and $K_{+-}$.
  • Figure 4: Three-dimensional illustrations of $\mathcal{T}'$ and $K_{+-}$.

Theorems & Definitions (33)

  • Definition 1.1
  • Remark 1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Remark 2
  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • ...and 23 more