  A framework to systematically decouple high order elliptic equations into
combination of Poisson-type and Stokes-type equations is developed. The key is
to systematically construct the underling commutative diagrams involving the
complexes and Helmholtz decompositions in a general way. Discretizing the
decoupled formulation leads to a natural superconvergence between the Galerkin
projection and the decoupled approximation. Examples include but not limit to:
the primal formulations and mixed formulations of biharmonic equation, fourth
order curl equation, and triharmonic equation etc. As a by-product, Helmholtz
decompositions for many dual spaces are obtained.
