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 ${\mathbf{D}}_{\mathbf{f}}$ 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:

$$\varphi =\mathrm{\nabla }\cdot \overrightarrow{u}=\frac{\partial u_x}{\partial x}+\frac{\partial u_y}{\partial y}+\frac{\partial u_z}{\partial z}$$

Where $\mathbf{u}$ is the discrete representation of the continuous variable $\overrightarrow{u}$ on cell faces and $\mathit{\varphi }$ is the discrete representation of $\varphi$ at cell centers, face_divergence constructs a discrete linear operator ${\mathbf{D}}_{\mathbf{f}}$ such that:

$$\mathit{\varphi }={\mathbf{D}}_{\mathbf{f}}\mathbf{u}$$

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:

$${\varphi }_i=\frac{1}{V_i}\sum _{k\in K}{A}_k{\overrightarrow{u}}_k\cdot {\hat{n}}_k$$

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