discretize.TreeMesh.average_face_to_cell#

TreeMesh.average_face_to_cell#

Averaging operator from faces to cell centers (scalar quantities).

This property constructs a 2nd order averaging operator that maps scalar quantities from faces to cell centers. This averaging operator is used when a discrete scalar quantity defined on mesh faces must be projected to cell centers. Once constructed, the operator is stored permanently as a property of the mesh. See notes.

Returns:

(n_cells, n_faces) scipy.sparse.csr_matrix

The scalar averaging operator from faces to cell centers

Notes

Let ${\mathit{\varphi }}_{\mathit{f}}$ be a discrete scalar quantity that lives on mesh faces. average_face_to_cell constructs a discrete linear operator ${\mathbf{A}}_{\mathbf{fc}}$ that projects ${\mathit{\varphi }}_{\mathit{f}}$ to cell centers, i.e.:

$${\mathit{\varphi }}_{\mathit{c}}={\mathbf{A}}_{\mathbf{fc}}{\mathit{\varphi }}_{\mathit{f}}$$

where ${\mathit{\varphi }}_{\mathit{c}}$ approximates the value of the scalar quantity at cell centers. For each cell, we are simply averaging the values defined on its faces. The operation is implemented as a matrix vector product, i.e.:

phi_c = Afc @ phi_f