discretize.SimplexMesh.boundary_node_vector_integral

property SimplexMesh.boundary_node_vector_integral

Represent the operation of integrating a vector function dotted with the boundary normal.

This matrix represents the boundary surface integral of a vector function dotted with the boundary normal and multiplied with a scalar finite volume test function on the mesh.

Returns:

(n_nodes, dim * n_boundary_nodes) scipy.sparse.csr_matrix

Sparse matrix of shape.

Notes

The integral we are representing on the boundary of the mesh is

\[\int_{\Omega} (w \vec{u}) \cdot \hat{n} \partial \Omega\]

In discrete form this is:

\[w^T * P @ u_b\]

where w is defined on all nodes, and u_b is all three components defined on boundary nodes.