Module ciarlet 

Ciarlet finite elements.

In the Ciarlet definition, a finite element is defined to be a triple $(\mathcal{R}, \mathcal{V}, \mathcal{L})$, where:

• $\mathcal{R}\subset\mathbb{R}^d$ is the reference cell,
• $\mathcal{V}$ is a finite dimensional function space on $\mathcal{R}$ of dimension $n$, usually polynomials,
• $\mathcal{L} = {\ell_0,\dots, \ell_{n-1}}$ is a basis of the dual space $\mathcal{V}^* = \set{f:\mathcal{V} -> \mathbb{R}: f\text{ is linear}}$, with

The basis functions $\phi_0,\dots, \phi_{n-1}$ of the finite element space are defined by $$\ell_i(\phi_j) = \begin{cases}1 &\text{if }~i = j \newline 0 &\text{otherwise}\end{cases}. $$

Example

The order 1 Lagrange space on a triangle is defined by:

• $\mathcal{R}$ is a triangle with vertices $(0, 0)$, $(1, 0)$, $(0, 1)$.
• $\mathcal{V} = \text{span}\set{1, x, y}$.
• $\mathcal{L} = \set{\ell_0, \ell_0, \ell_1}$ with $\ell_j$ the pointwise evaluation at vertex $v_j$, and each functional associated with the relevant vertex.

This gives the basis functions $\phi_0(x, y) = 1 - x - y$, $\phi_1(x, y) = x$, $\phi_2(x, y) = y$.

References

• https://defelement.org/ciarlet.html

Re-exports

pub use lagrange::LagrangeElementFamily;

pub use nedelec::NedelecFirstKindElementFamily;

pub use raviart_thomas::RaviartThomasElementFamily;

Modules

lagrange

Lagrange elements.

nedelec

Nedelec elements.

raviart_thomas

Raviart-Thomas elements.

Structs

CiarletElement

A Ciarlet element