  In this article, discrete variants of several results from vector calculus
are studied for classical finite difference summation by parts operators in two
and three space dimensions. It is shown that existence theorems for
scalar/vector potentials of irrotational/solenoidal vector fields cannot hold
discretely because of grid oscillations, which are characterised explicitly.
This results in a non-vanishing remainder associated to grid oscillations in
the discrete Helmholtz Hodge decomposition. Nevertheless, iterative numerical
methods based on an interpretation of the Helmholtz Hodge decomposition via
orthogonal projections are proposed and applied successfully. In numerical
experiments, the discrete remainder vanishes and the potentials converge with
the same order of accuracy as usual in other first order partial differential
equations. Motivated by the successful application of the Helmholtz Hodge
decomposition in theoretical plasma physics, applications to the discrete
analysis of magnetohydrodynamic (MHD) wave modes are presented and discussed.
