A framework to systematically construct differential complex and Helmholtz decompositions is developed. The Helmholtz decomposition is used to decouple the mixed formulation of high order elliptic equations into combination of Poisson-type and Stokes-type equations. By finding the underlying complex, this decomposition is applied in the discretization level to design fast solvers for solving the linear algebraic system. It can be also applied in the continuous level first and then discretize the decoupled formulation, which leads to a natural superconvergence between the Galerkin projection and the decoupled approximation. Examples include but are not limited to: biharmonic equation, triharmonic equation, fourth order curl equation, HHJ mixed method for plate problem, and Reissner-Mindlin plate model etc. As a by-product, Helmholtz decompositions for many dual spaces are obtained. This is a joint work with Prof. Long Chen from UCI.