discretize.TreeMesh.edge_curl#

TreeMesh.edge_curl#

Edge curl operator (edges to faces)

This property constructs the 2nd order numerical curl operator that maps from edges to faces. The operator is a sparse matrix ${\mathbf{C}}_{\mathbf{e}}$ that can be applied as a matrix-vector product to a discrete vector quantity u that lives on the edges; i.e.:

curl_u = Ce @ u

Once constructed, the operator is stored permanently as a property of the mesh.

Returns:

(n_faces, n_edges) scipy.sparse.csr_matrix

The numerical curl operator from edges to faces

Notes

In continuous space, the curl operator is defined as:

$$\begin{array}{r}\overrightarrow{w}=\mathrm{\nabla }\times \overrightarrow{u}=\left|\begin{array}{ccc}\hat{x}& \hat{y}& \hat{z}\\ {\partial }_x& {\partial }_y& {\partial }_z\\ u_x& u_y& u_z\end{array}\right|\end{array}$$

Where $\mathbf{u}$ is the discrete representation of the continuous variable $\overrightarrow{u}$ on cell edges and $\mathbf{w}$ is the discrete representation of the curl on the faces, edge_curl constructs a discrete linear operator ${\mathbf{C}}_{\mathbf{e}}$ such that:

$$\mathbf{w}={\mathbf{C}}_{\mathbf{e}}\mathbf{u}$$

The computation of the curl on mesh faces can be expressed according to the integral form below. For face $i$ bordered by a set of edges indexed by subset $K$:

$$w_i=\frac{1}{A_i}\sum _{k\in K}{\overrightarrow{u}}_k\cdot {\overrightarrow{\ell }}_k$$

where $A_i$ is the surface area of face i, $u_k$ is the value of $\overrightarrow{u}$ on face k, and vec{ell}_k is the path along edge k.