Dohp dohp

The natural ordering to start with has global dofs essentially sorted as [vertices, edges, faces, regions]. While this is a fine ordering for matrix-free application, it is not good when using an assembled matrix in the preconditioner (mostly we're thinking of SOR, ILU(k) and LU, perhaps as part of a multigrid scheme). A low-bandwidth ordering (different from a fill-reducing ordering) should give better cache performance.

It is natural to compute this ordering while creating the function space, so that subsequent assembly can use the good ordering. Obviously the corresponding scalar function space is sufficient for computing the reordering, but it might be possible to make it cheaper while keeping all interior dofs for each region together. Suppose we compute an ordering by assigning one dof to all the interior nodes of every entity (with a positive number of nodes). For any space of at least quadratic elements, the reordering would be computed by assembling the nonzero pattern for a matrix corresponding to the Q1 basis induced by quadratic elements. For order 5, this corresponds to more than a factor of 10 savings (i.e. the ordering should be very cheap to compute).