On the kernel of the discrete gradient operator

When solving the Stokes problem on semistaggered grids, a discrete analog of the gradient operator with a nontrivial kernel arises. This fact may lead to loss of accuracy in a discrete solution and to some difficulties in the iterative solution of the problem. One of the approaches to the construction of efficient numerical methods on semistaggered grids is based on the assumption that the structure of the gradient kernel is known. In this paper, a system of linearly independent functions from the discrete gradient kernel is constructed for the two- and three-dimensional cases. The numerical results allow one to assume that the resulting system forms the kernel basis.