discretize.TreeMesh.average_edge_to_face_vector

TreeMesh.average_edge_to_face_vector

Averaging operator from edges to faces (vector quantities).

This property constructs the averaging operator that independently maps the Cartesian components of vector quantities from edges to faces. This averaging operators is used when a discrete vector 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 vector averaging operator from edges to faces.

Notes

Let \(\mathbf{u_e}\) be the discrete representation of a vector quantity whose Cartesian components are defined on their respective edges; e.g. the x-component is defined on x-edges. average_edge_to_face_vector constructs a discrete linear operator \(\mathbf{A_{ef}}\) that projects each Cartesian component of \(\mathbf{u_e}\) to its corresponding face, i.e.:

\[\mathbf{u_f} = \mathbf{A_{ef}} \, \mathbf{u_e}\]

where \(\mathbf{u_f}\) is a discrete vector quantity whose Cartesian components are defined on their respective faces; e.g. the x-component is defined on x-faces.