Fast orthogonalization to the kernel of the discrete gradient operator with application to Stokes problem

We obtain a simple tensor representation of the kernel of the discrete d-dimensional gradient operator defined on tensor semi-staggered grids. We show that the dimension of the nullspace grows as O(n^d-2), where d is the dimension of the problem, and n is one-dimensional grid size. The tensor structure allows fast orthogonalization to the kernel. The usefulness of such procedure is demonstrated on three-dimensional Stokes problem, discretized by finite differences on semi-staggered grids, and it is shown by numerical experiments that the new method outperforms usually used stabilization approach.