(Note: this post is mainly for me to consolidate my thoughts)

In the framework of domain decomposition, consider creating the Schur complement which orthogonalizes interior nodes and the edges/vertex nodes. It turns out the norm of these functions which are orthogonal to the interior functions are minimal energy (i.e. L2 norm) extensions.

This can be seen in both a Hilbert space way or an optimization way. For the optimization way, write out the product for the mass matrix, and note that we can take a derivative to minimize one of the factors… now the Schur complement pops up naturally!