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 \(\mathbf{C_e}\) 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:

\[\begin{split}\vec{w} = \nabla \times \vec{u} = \begin{vmatrix} \hat{x} & \hat{y} & \hat{z} \\ \partial_x & \partial_y & \partial_z \\ u_x & u_y & u_z \end{vmatrix}\end{split}\]

Where \(\mathbf{u}\) is the discrete representation of the continuous variable \(\vec{u}\) on cell edges and \(\mathbf{w}\) is the discrete representation of the curl on the faces, edge_curl constructs a discrete linear operator \(\\mathbf{C_e}\) such that:

\[\mathbf{w} = \mathbf{C_e \, u}\]

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 \(K\):

\[w_i = \frac{1}{A_i} \sum_{k \in K} \vec{u}_k \cdot \vec{\ell}_k\]

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