  Recent work on octree-based finite-element systems has developed a multigrid
solver for Poisson equations on meshes. While the idea of defining a regularly
indexed function space has been successfully used in a number of applications,
it has also been noted that the richness of the function space is limited
because the function values can be coupled across locally disconnected regions.
In this work, we show how to enrich the function space by introducing functions
that resolve the coupling while still preserving the nesting hierarchy that
supports multigrid. A spectral analysis reveals the superior quality of the
resulting Laplace-Beltrami operator and applications to surface flow
demonstrate that our new solver more efficiently converges to the correct
solution.
