discretize.operators.InnerProducts.average_edge_to_face

property InnerProducts.average_edge_to_face

Averaging operator from edges to faces.

This property constructs the averaging operator that maps uantities from edges to faces. This averaging operators is used when a discrete 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 averaging operator from edges to faces.

Notes

Let \(\mathbf{u_e}\) be the discrete representation of aquantity whose that is defined on the edges. average_edge_to_face constructs a discrete linear operator \(\mathbf{A_{ef}}\) that projects \(\mathbf{u_e}\) to its corresponding face, i.e.:

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

where \(\mathbf{u_f}\) is a quantity defined on the respective faces.

Examples

Here we compute the values of a vector function discretized to the edges. We then create an averaging operator to approximate the function on the faces.

We start by importing the necessary packages and defining a mesh.

>>> from discretize import TensorMesh
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> h = 0.5 * np.ones(40)
>>> mesh = TensorMesh([h, h], x0="CC")

Create a discrete vector on mesh edges

>>> edges = mesh.edges
>>> u_ex = -(edges[:, 1] / np.sqrt(np.sum(edges ** 2, axis=1))) * np.exp(
...     -(edges[:, 0] ** 2 + edges[:, 1] ** 2) / 6 ** 2
... )
>>> u_ey = (edges[:, 0] / np.sqrt(np.sum(edges ** 2, axis=1))) * np.exp(
...     -(edges[:, 0] ** 2 + edges[:, 1] ** 2) / 6 ** 2
... )
>>> u_e = np.c_[u_ex, u_ey]

Next, we construct the averaging operator and apply it to the discrete vector quantity to approximate the value at the faces.

>>> Aef = mesh.average_edge_to_face
>>> u_f = Aef @ u_e

Plot the results,

>>> fig = plt.figure(figsize=(11, 5))
>>> ax1 = fig.add_subplot(121)
>>> proj_ue = mesh.project_edge_vector(u_e)
>>> mesh.plot_image(proj_ue, ax=ax1, v_type="E", view='vec')
>>> ax1.set_title("Variable at edges", fontsize=16)
>>> ax2 = fig.add_subplot(122)
>>> proj_uf = mesh.project_face_vector(u_f)
>>> mesh.plot_image(proj_uf, ax=ax2, v_type="F", view='vec')
>>> ax2.set_title("Averaged to faces", fontsize=16)
>>> plt.show()

(Source code, png, pdf)

Below, we show a spy plot illustrating the sparsity and mapping of the operator

>>> fig = plt.figure(figsize=(9, 9))
>>> ax1 = fig.add_subplot(111)
>>> ax1.spy(Aef, ms=1)
>>> ax1.set_title("Edge Index", fontsize=12, pad=5)
>>> ax1.set_ylabel("Face Index", fontsize=12)
>>> plt.show()

(png, pdf)