  We introduce a geometric multigrid method for solving linear systems arising
from variational problems on surfaces in geometry processing, Gravo MG. Our
scheme uses point clouds as a reduced representation of the levels of the
multigrid hierarchy to achieve a fast hierarchy construction and to extend the
applicability of the method from triangle meshes to other surface
representations like point clouds, nonmanifold meshes, and polygonal meshes. To
build the prolongation operators, we associate each point of the hierarchy to a
triangle constructed from points in the next coarser level. We obtain
well-shaped candidate triangles by computing graph Voronoi diagrams centered
around the coarse points and determining neighboring Voronoi cells. Our
selection of triangles ensures that the connections of each point to points at
adjacent coarser and finer levels are balanced in the tangential directions. As
a result, we obtain sparse prolongation matrices with three entries per row and
fast convergence of the solver.
