  The main difficulty in solving the discrete source or eigenvalue problems of
the operator $ d^*d $ with iterative methods is to deal with its huge kernel,
for example, the $ \nabla \times \nabla \times $ and $- \nabla ( \nabla \cdot )
$ operator. In this paper, we construct a kind of auxiliary schemes for their
discrete systems based on Hodge Laplacian on de Rham complex. The spectra of
the new schemes are Laplace-like. Then many efficient iterative methods and
preconditioning techniques can be applied to them. After getting the solutions
of the auxiliary schemes, the desired solutions of the original systems can be
recovered or recognized through some simple operations. We sum these up as a
new framework to compute the discrete source and eigenvalue problems of the
operator $ d^*d $ using iterative method. We also investigate two
preconditioners for the auxiliary schemes, ILU-type method and Multigrid
method. Finally, we present plenty of numerical experiments to verify the
efficiency of the auxiliary schemes.
