We develop a preconditioner and an iterative method for the linear system resulting from the discretization of second order elliptic problems by the symmetric discontinuous Galerkin methods. The key is a new stable decomposition of the piecewise polynomial discontinuous finite element space into a conforming space and a space containing the high frequency components. By using this new decomposition, we then prove that both the condition numbers of the preconditioner and the convergent rate of the iterative method are independent of the mesh size. Numerical experiments are also shown to confirm these theoretical results.