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 ${\mathbf{u}}_{\mathbf{e}}$ be the discrete representation of aquantity whose that is defined on the edges. average_edge_to_face constructs a discrete linear operator ${\mathbf{A}}_{\mathbf{ef}}$ that projects ${\mathbf{u}}_{\mathbf{e}}$ to its corresponding face, i.e.:

$${\mathbf{u}}_{\mathbf{f}}={\mathbf{A}}_{\mathbf{ef}}{\mathbf{u}}_{\mathbf{e}}$$

where ${\mathbf{u}}_{\mathbf{f}}$ is a quantity defined on the respective faces.