# discretize.TreeMesh.average_face_x_to_cell#

TreeMesh.average_face_x_to_cell#

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

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

Returns
(n_cells, n_faces_x) scipy.sparse.csr_matrix

The scalar averaging operator from x-faces to cell centers

Notes

Let $$\boldsymbol{\phi_x}$$ be a discrete scalar quantity that lives on x-faces. average_face_x_to_cell constructs a discrete linear operator $$\mathbf{A_{xc}}$$ that projects $$\boldsymbol{\phi_x}$$ to cell centers, i.e.:

$\boldsymbol{\phi_c} = \mathbf{A_{xc}} \, \boldsymbol{\phi_x}$

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

phi_c = Axc @ phi_x