# number of tetrahedra in unstructured mesh when edge length is halved

For a given 3D domain, two unstructured tetrahedral meshes M_1 and M_2 are generated. The average edge length in M_1 is double the average edge length of M_2.

Are there estimates on the ratio of the number of tetrahedra in both meshes?

• Wouldn't it simply be ~8x? – stali Jun 30 '13 at 23:26
• @stali Adaptive meshes could have a factor less than that. – Shuhao Cao Jul 1 '13 at 4:05
• @stali That would be true for hexahedral meshes. – Nico Schlömer Jul 1 '13 at 7:58
• Why dont you quickly test in Cubit/Gmsh? – stali Jul 1 '13 at 13:13
• @stali I played around a little with Gmsh, and found that when the average edge length is scaled by a factor of 0.5, the number of tetrahedra increases by a factor of about 6. I don't know under which conditions this holds though. – Nico Schlömer Jul 1 '13 at 18:38

If the edge lengths decrease by a factor $k$, the number of tetrahedra will typically increase by between $k^2$ and $k^3$ depending on the adaptivity of the mesh. A purely uniform mesh will see $k^3$, just as a uniform grid of length $1/n$ has $O(n^3)$ cells. A highly adaptive mesh with full resolution near a surface will increase by roughly $k^2$, since most of the elements will occur in a band near the surface. It is also possible to see only a $k$ factor increase if the refinement is near a curve, but this is less common.