discretize.base.BaseRegularMesh.nodal_laplacian#

property BaseRegularMesh.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 ${\mathbf{L}}_{\mathbf{n}}$ that can be applied as a matrix-vector product to a discrete scalar quantity $\mathit{\varphi }$ that lives on the nodes, i.e.:

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:

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

Where $\mathit{\varphi }$ is the discrete representation of the continuous variable $\varphi$ on the nodes, and $\mathit{\psi }$ is the discrete representation of its scalar Laplacian on the nodes, nodal_laplacian constructs a discrete linear operator ${\mathbf{L}}_{\mathbf{n}}$ such that:

$$\mathit{\psi }={\mathbf{L}}_{\mathbf{n}}\mathit{\varphi }$$