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 $$\mathbf{G_n}$$ that can be applied as a matrix-vector product to a discrete scalar quantity $$\boldsymbol{\phi}$$ 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:

$\vec{u} = \nabla \phi = \frac{\partial \phi}{\partial x}\hat{x} + \frac{\partial \phi}{\partial y}\hat{y} + \frac{\partial \phi}{\partial z}\hat{z}$

Where $$\boldsymbol{\phi}$$ is the discrete representation of the continuous variable $$\phi$$ on the nodes and $$\mathbf{u}$$ is the discrete representation of $$\vec{u}$$ on the edges, nodal_gradient constructs a discrete linear operator $$\mathbf{G_n}$$ such that:

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

The Cartesian components of $$\vec{u}$$ 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 $$n_2$$:

$u_i = \frac{\phi_{n_2} - \phi_{n_1}}{h_i}$

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