Numerous physical systems, such as magnetohydrodynamics and electromagnetics to name a few, rely on the evolution of vector fields. The vector fields have the special property that if the
vector field is divergence-free at one point of time (which it is) then it will remain divergence-free for all later times. It is well-known that the build up of numerically generated divergence
results in a loss of physical consistency. Such systems rely on a Stokes-law type of update equation. It is shown that the update can be formulated so that the vector field  remains divergence-free.

      Further challenges arise when trying to do adaptive mesh refinement (AMR) for such systems. The process of carrying out divergence-free prolongation across meshes forces one to make fundamental innovations in the divergence-free reconstruction of vector fields. It is shown that by the use of such innovations, one can arrive at an AMR strategy for MHD that is truly divergence-free. It will also be shown that these innovations carry over to triangulated (and tetrahedral) meshes. The innovations also yield new, interesting, multidimensional schemes
for MHD.