discretize.TreeMesh.average_node_to_face#

TreeMesh.average_node_to_face#

Averaging operator from nodes to faces (scalar quantities).

This property constructs a 2nd order averaging operator that maps scalar quantities from nodes to edges; scalar at faces is organized in a 1D numpy.array of the form [x-faces, y-faces, z-faces]. This averaging operator is used when a discrete scalar quantity defined on mesh nodes must be projected to faces. Once constructed, the operator is stored permanently as a property of the mesh. See notes.

Returns:

(n_faces, n_nodes) scipy.sparse.csr_matrix

The scalar averaging operator from nodes to faces

Notes

Let ${\mathit{\varphi }}_{\mathit{n}}$ be a discrete scalar quantity that lives on mesh nodes. average_node_to_face constructs a discrete linear operator ${\mathbf{A}}_{\mathbf{nf}}$ that projects ${\mathit{\varphi }}_{\mathit{n}}$ to faces, i.e.:

$${\mathit{\varphi }}_{\mathit{f}}={\mathbf{A}}_{\mathbf{nf}}{\mathit{\varphi }}_{\mathit{n}}$$

where ${\mathit{\varphi }}_{\mathit{f}}$ approximates the value of the scalar quantity at faces. For each face, we are simply averaging the values at the nodes which outline the face. The operation is implemented as a matrix vector product, i.e.:

phi_f = Anf @ phi_n