In the algorighm the two endpoints of an edge are collapsed onto the midpoint of the edge, removing all the elements that were formed with that edge and enlarging all the elements that are formed with one of the endpoints of the edge. This process is applied sequentially to a set of colored edges until a prescribed coarsening ratio is achieved. In this graph, any edge can be collapsed if after the collapse the geometry is still valid and none of the neighboring edges exceeds a maximum length it has been allowed. The first criterion is obvious, we cannot tolerate negative volumes due to folded grids. The second criterion expresses the design principle of multigrid, usually the coarse grid doubles the length of the edges in a mesh.
Edges are collapsed in order of their length, the shortest edge is tested for a possible collapse.
Special care has to be taken to achieve directional coarsening in highly stretched meshes. However, the 3D extension of this edge-based collapsing algorithm to coarsen meshes with tetrahedra, pyramids, prims and hexahedra proved unsuccessful. The 2D algorithm attempts to collapse one edge after the other, in the order of shortest length. While a tetrahedron with a collapsed edge vanishes, all other hybrid element types see their volume reduced. Collapsing on edges created coarser grids with a large number of elements of small volume.