Degrees of Freedom (DOFs)¶

An abstract finite element is defined by:

A reference element \(R\subset\mathbb{R}^d\). In Bempp, \(R\) is always a triangle with vertices at \((0,0)\), \((1,0)\) and \((0,1)\).
A finite dimensional polynomial space \(\mathcal{V}\). Inside each triangle in the mesh, the solution will be approximated by a function in this space.
A set of functionals \(\mathcal{L}={f_1, ... f_n}\) that form a basis of the dual space \(\mathcal{V}^*=\{f:\mathcal{V}\to\mathbb{R}\}\).

Given a functional \(f_i\in\mathcal{L}\), a corresponding polynomial basis function \(\phi_i\in\mathcal{V}\) is defined as the function such that

\[\begin{split} f_j(\phi_i)=\begin{cases}1&i=j\\0&i\not=j\end{cases}. \end{split}\]

Example: P1 space¶

As an example, for a P1 (continuous piecewise linear space) the following are used:

In this case, it is common to say that the space has a DOF at each vertex of the mesh.

The definitions of the spaces available in Bempp are summarised in the following table. In each case, \(R\) is the unit triangle.

Space	\(\mathcal{V}\)	\(\mathcal{L}\)
DP0	\(\operatorname{span}\{1\}\)	Point evaluation at centre of \(R\)
P1	\(\operatorname{span}\{1, x, y\}\)	point evaluations at vertices of \(R\)
RWG1	\(\operatorname{span}\left\{\left(\begin{array}{c}1\\0\end{array}\right),\left(\begin{array}{c}0\\1\end{array}\right),\left(\begin{array}{c}x\\y\end{array}\right)\right\}\)	Point evaluations at the midpoints of edges of \(R\) in a direction normal to the edge
SNC1	\(\operatorname{span}\left\{\left(\begin{array}{c}1\\0\end{array}\right),\left(\begin{array}{c}0\\1\end{array}\right),\left(\begin{array}{c}y\\-x\end{array}\right)\right\}\)	Point evaluations at the midpoints of edges of \(R\) in a direction tangential to the edge

The spaces defined on the barycentric dual grid are defined as subspaces of the spaces in the table above. Their definitions can be found in A dual finite element complex on the barycentric refinement (2007) by A. Buffa and S. Christiansen.