# discretize.TreeMesh.face_divergence#

TreeMesh.face_divergence#

Face divergence operator (faces to cell-centres).

This property constructs the 2nd order numerical divergence operator that maps from faces to cell centers. The operator is a sparse matrix $$\mathbf{D_f}$$ that can be applied as a matrix-vector product to a discrete vector $$\mathbf{u}$$ that lives on mesh faces; i.e.:

div_u = Df @ u


Once constructed, the operator is stored permanently as a property of the mesh. See notes for additional details.

Returns
(n_cells, n_faces) scipy.sparse.csr_matrix

The numerical divergence operator from faces to cell centers

Notes

In continuous space, the divergence operator is defined as:

$\phi = \nabla \cdot \vec{u} = \frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} + \frac{\partial u_z}{\partial z}$

Where $$\mathbf{u}$$ is the discrete representation of the continuous variable $$\vec{u}$$ on cell faces and $$\boldsymbol{\phi}$$ is the discrete representation of $$\phi$$ at cell centers, face_divergence constructs a discrete linear operator $$\mathbf{D_f}$$ such that:

$\boldsymbol{\phi} = \mathbf{D_f \, u}$

For each cell, the computation of the face divergence can be expressed according to the integral form below. For cell $$i$$ whose corresponding faces are indexed as a subset $$K$$ from the set of all mesh faces:

$\phi_i = \frac{1}{V_i} \sum_{k \in K} A_k \, \vec{u}_k \cdot \hat{n}_k$

where $$V_i$$ is the volume of cell $$i$$, $$A_k$$ is the surface area of face k, $$\vec{u}_k$$ is the value of $$\vec{u}$$ on face k, and $$\hat{n}_k$$ represents the outward normal vector of face k for cell i.