discretize.TreeMesh.nodal_gradient#

TreeMesh.nodal_gradient#

Nodal gradient operator (nodes to edges)

This property constructs the 2nd order numerical gradient operator that maps from nodes to edges. The operator is a sparse matrix ${\mathbf{G}}_{\mathbf{n}}$ that can be applied as a matrix-vector product to a discrete scalar quantity $\mathit{\varphi }$ that lives on the nodes, i.e.:

grad_phi = Gn @ phi

Once constructed, the operator is stored permanently as a property of the mesh.

Returns:

(n_edges, n_nodes) scipy.sparse.csr_matrix

The numerical gradient operator from nodes to edges

Notes

In continuous space, the gradient operator is defined as:

$$\overrightarrow{u}=\mathrm{\nabla }\varphi =\frac{\partial \varphi }{\partial x}\hat{x}+\frac{\partial \varphi }{\partial y}\hat{y}+\frac{\partial \varphi }{\partial z}\hat{z}$$

Where $\mathit{\varphi }$ is the discrete representation of the continuous variable $\varphi$ on the nodes and $\mathbf{u}$ is the discrete representation of $\overrightarrow{u}$ on the edges, nodal_gradient constructs a discrete linear operator ${\mathbf{G}}_{\mathbf{n}}$ such that:

$$\mathbf{u}={\mathbf{G}}_{\mathbf{n}}\mathit{\varphi }$$

The Cartesian components of $\overrightarrow{u}$ are defined on their corresponding edges (x, y or z) as follows; e.g. the x-component of the gradient is defined on x-edges. For edge $i$ which defines a straight path of length $h_i$ between adjacent nodes $n_1$ and $n_2$:

$$u_i=\frac{{\varphi }_{n_2}-{\varphi }_{n_1}}{h_i}$$

Note that $u_i\in \mathbf{u}$ may correspond to a value on an x, y or z edge. See the example below.