# discretize.TensorMesh.nodal_laplacian¶

property TensorMesh.nodal_laplacian

Nodal scalar Laplacian operator (nodes to nodes)

This property constructs the 2nd order scalar Laplacian operator that maps from nodes to nodes. The operator is a sparse matrix $$\mathbf{L_n}$$ that can be applied as a matrix-vector product to a discrete scalar quantity $$\boldsymbol{\phi}$$ that lives on the nodes, i.e.:

laplace_phi = Ln @ phi


The operator * assumes a zero Neumann boundary condition for the discrete scalar quantity. Once constructed, the operator is stored permanently as a property of the mesh.

Returns
(n_nodes, n_nodes) scipy.sparse.csr_matrix

The numerical Laplacian operator from nodes to nodes

Notes

In continuous space, the scalar Laplacian operator is defined as:

$\psi = \nabla^2 \phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2}$

Where $$\boldsymbol{\phi}$$ is the discrete representation of the continuous variable $$\phi$$ on the nodes, and $$\boldsymbol{\psi}$$ is the discrete representation of its scalar Laplacian on the nodes, nodal_laplacian constructs a discrete linear operator $$\mathbf{L_n}$$ such that:

$\boldsymbol{\psi} = \mathbf{L_n} \, \boldsymbol{\phi}$