  In this paper, we propose and analyze an efficient preconditioning method for
the elliptic problem based on the reconstructed discontinuous approximation
method. We reconstruct a high-order piecewise polynomial space that arbitrary
order can be achieved with one degree of freedom per element. This space can be
directly used with the symmetric/nonsymmetric interior penalty discontinuous
Galerkin method. Compared with the standard DG method, we can enjoy the
advantage on the efficiency of the approximation. Besides, we establish an norm
equivalence result between the reconstructed high-order space and the piecewise
constant space. This property further allows us to construct an optimal
preconditioner from the piecewise constant space. The upper bound of the
condition number to the preconditioned symmetric/nonsymmetric system is shown
to be independent of the mesh size. Numerical experiments are provided to
demonstrate the validity of the theory and the efficiency of the proposed
method.
