The symmetric discontinuous Galerkin method does not need stabilization in 1D for polynomial orders p ≥ 2

In this Note we prove that in one space dimension, the symmetric discontinuous Galerkin method for second order elliptic problems is stable for polynomial orders $p \ge 2$ without using any stabilization parameter. The method yields optimal convergence rates in both the broken energy norm and the $L^2$- norm and can be written in conservative form with fluxes independent of any stabilization parameter.