discretize.TreeMesh.average_edge_to_face

TreeMesh.average_edge_to_face

Averaging operator from edges to faces.

This property constructs the averaging operator that maps uantities from edges to faces. This averaging operators is used when a discrete quantity defined on mesh edges must be approximated at faces. The operation is implemented as a matrix vector product, i.e.:

u_f = Aef @ u_e

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

Returns:

(n_faces, n_edges) scipy.sparse.csr_matrix

The averaging operator from edges to faces.

Notes

Let \(\mathbf{u_e}\) be the discrete representation of aquantity whose that is defined on the edges. average_edge_to_face constructs a discrete linear operator \(\mathbf{A_{ef}}\) that projects \(\mathbf{u_e}\) to its corresponding face, i.e.:

\[\mathbf{u_f} = \mathbf{A_{ef}} \, \mathbf{u_e}\]

where \(\mathbf{u_f}\) is a quantity defined on the respective faces.