discretize.operators.DiffOperators.average_edge_to_face_vector

property DiffOperators.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.

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_x = mesh.edges_x
>>> edges_y = mesh.edges_y
>>> u_ex = -(edges_x[:, 1] / np.sqrt(np.sum(edges_x ** 2, axis=1))) * np.exp(
...     -(edges_x[:, 0] ** 2 + edges_x[:, 1] ** 2) / 6 ** 2
... )
>>> u_ey = (edges_y[:, 0] / np.sqrt(np.sum(edges_y ** 2, axis=1))) * np.exp(
...     -(edges_y[:, 0] ** 2 + edges_y[:, 1] ** 2) / 6 ** 2
... )
>>> u_e = np.r_[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_vector
>>> u_f = Aef @ u_e

Plot the results,

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

(Source code, png, pdf)

../../_images/discretize-operators-DiffOperators-average_edge_to_face_vector-1_00_00.png

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

Expand to show scripting for plot
>>> 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)

../../_images/discretize-operators-DiffOperators-average_edge_to_face_vector-1_01_00.png