Size complexity of volume meshes vs. surface meshes

Typical volume meshes in three dimensions are designed to conform to an underlying two-dimensional surface mesh, with volume mesh element size growing larger away from the surface. The surface mesh may be uni-formly spaced or highly graded, and may have fine resolu-tion due to extrinsic mesh size concerns. When we desire that such a volume mesh have good aspect ratio, we re-quire that some space-filling scaffold vertices be inserted off the surface. We analyze the number of scaffold ver-tices in a setting that encompasses many existing volume meshing algorithms. We show that under simple precon-ditions, the number of scaffold vertices will be linear in the number of surface vertices. 1