discretize.TreeMesh.average_node_to_edge#

TreeMesh.average_node_to_edge#

Averaging operator from nodes to edges (scalar quantities).

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

Returns:

(n_edges, n_nodes) scipy.sparse.csr_matrix

The scalar averaging operator from nodes to edges

Notes

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

$${\mathit{\varphi }}_{\mathit{e}}={\mathbf{A}}_{\mathbf{ne}}{\mathit{\varphi }}_{\mathit{n}}$$

where ${\mathit{\varphi }}_{\mathit{e}}$ approximates the value of the scalar quantity at edges. For each edge, we are simply averaging the values defined on the nodes it connects. The operation is implemented as a matrix vector product, i.e.:

phi_e = Ane @ phi_n