General concepts » Ciarlet’s definition of a finite element

The Ciarlet triplet $ (K, P, \mathcal{N}) $ .

Introduction

In the finite element method, each “element” is described by a reference geometric polytope, a space of local shape functions on that polytope, and a unisolvent set of degrees of freedom. The classical formulation is due to Ciarlet.

Ciarlet’s Definition

A finite element is the triple

\[ (K, P, \mathcal{N}) \]

where:

  • $ K $ is the reference polytope, a closed, bounded convex polytope in $ \mathbb{R}^d $ with nonempty interior.
  • $ P $ is the space of shape functions, a finite-dimensional subspace of $ C^0(K) $ , typically a polynomial space $ \mathbb{P}_k(K) $ .

  • $ \mathcal{N} $ is the set of degrees of freedom, a unisolvent family of linear functionals $ \ell_i : P \to \mathbb{R} $ satisfying

    \[ \bigl[\,p \in P,\; \ell_i(p)=0\;\forall\,i\,\bigr] \;\Longrightarrow\; p \equiv 0. \]

Unisolvence of $ \mathcal{N} $ means the evaluation map

\[ P \;\longrightarrow\; \mathbb{R}^N, \quad p \;\mapsto\; \bigl(\ell_1(p),\dots,\ell_N(p)\bigr) \]

is an isomorphism.

Local Interpolation Operator

Given a Ciarlet element $(K,P,\mathcal{N})$ with $\mathcal{N}=\{\ell_1,\dots,\ell_N\}$ , let $\{\varphi_1,\dots,\varphi_N\}\subset P$ be its dual basis: $\ell_j(\varphi_i)=\delta_{ij}$ . Then the local interpolation operator

\[ \mathcal{I} : C^0(K)\;\longrightarrow\;P, \quad \mathcal{I}[v] = \sum_{i=1}^N \ell_i(v)\,\varphi_i, \]

satisfies for all $v\in C^0(K)$ and all $i$ ,

\[ \ell_i\bigl(\mathcal{I}[v]\bigr) = \ell_i(v), \quad \mathcal{I}[p]=p\ \forall p\in P. \]

Lagrange @f$P_k@f$ Elements

On a reference polytope $K$ , the degree- $k$ Lagrange element is the Ciarlet triple

\[ (K,\;P_k(K),\;\mathcal{N}_k)\,, \]

where:

  • $P_k(K)=\mathbb{P}_k(K)$ , the space of scalar polynomials of total degree ≤ $k$ .
  • $\mathcal{N}_k=\{\ell_a\}_{a=1}^N$ , with each $\ell_a(p)=p(\xi_a)$ evaluation at the Lagrange node $\xi_a\in K$ .

When assembling over a mesh, continuity is enforced by gluing all local basis functions that share the same global node on:

  • Vertices (for $ k \ge 1 $ )
  • Edge-nodes (for $k \ge 2 $ )
  • Face-nodes (for $k \ge 3 $ in 3D)
  • Interior-nodes (for higher $k$ )

yielding a globally $C^0$ -conforming piecewise- $P_k$ space.

Raviart–Thomas @f$RT_k@f$ Elements

On $K$ , the Raviart–Thomas element of order $k$ is

\[ (K,\;P_{RT_k}(K),\;\mathcal{N}_{RT_k}) \]

with:

  • $P_{RT_k}(K)=\mathbb{P}_k(K)^d\oplus\{\xi\,q(\xi):q\in\mathbb{P}_k(K)\}$ .
  • $\mathcal{N}_{RT_k}$ given by

    • For each facet $F$ and all $q\in\mathbb{P}_k(F)$ ,

      \[ \ell_{F,q}(\mathbf{v}) = \int_F (\mathbf{v}\cdot\mathbf{n})\,q \,dS. \]

    • If $k\ge1$ , for each $\mathbf{r}\in\mathbb{P}_{k-1}(K)^d$ ,

      \[ \ell_{K,\mathbf{r}}(\mathbf{v}) = \int_K \mathbf{v}\cdot\mathbf{r} \,dV. \]

These DOFs enforce normal‐continuity across faces, yielding an $H(\mathrm{div})$ –conforming global space.

Nédélec (First Kind) @f$ND_k@f$ Elements

On $K$ , the Nédélec edge element of order $k$ (first kind) is

\[ (K,\;P_{ND_k}(K),\;\mathcal{N}_{ND_k}) \]

with:

  • $P_{ND_k}(K) = \mathbb{P}_k(K)^d \oplus \{\xi\times\mathbf{q}(\xi):\mathbf{q}\in\mathbb{P}_k(K)^d\}$ .
  • $\mathcal{N}_{ND_k}$ given by

    • For each edge $E$ and all $q\in\mathbb{P}_k(E)$ ,

      \[ \ell_{E,q}(\mathbf{v}) = \int_E (\mathbf{v}\cdot\mathbf{t})\,q \,ds. \]

    • If $k\ge1$ , for each face $F$ and all $\mathbf{r}\in\mathbb{P}_{k-1}(F)^d$ ,

      \[ \ell_{F,\mathbf{r}}(\mathbf{v}) = \int_F (\mathbf{n}\times\mathbf{v})\cdot\mathbf{r} \,dS. \]

These DOFs enforce tangential‐continuity across edges, yielding an $H(\mathrm{curl})$ –conforming global space.

« Connectivity | General concepts