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.