The discretizations of the derivative by the continuous Galerkin and the discontinuous Galerkin methods are exactly the same

Abstract: In the framework of ODEs, we uncover a new link between the continuous Galerkin method (see Math. Comp. (1972), 26 (118 and 120), 415-426 and 881-891) and the discontinuous Galerkin method (see Mathematical Aspects of Finite elements in PDEs, (1974), 89-123), namely, that the discretizations of the derivative by these two methods are the same. A direct consequence of this result is the construction of a new elementwise post-processing of the approximate solution provided by the Discontinuous Galerkin method. When the DG method uses polynomials of degree $k\ge0$, the post-processing consists in adding, to the DG approximate solution, the (scaled) left-Radau polynomial of degree $k+1$ multiplied by the jump of the approximate solution at the left boundary of the interval. No extra computation is required. The resulting new approximation is continuous and, for $k>0$, converges with order $k+2$, that is, with one order more than the original discontinuous Galerkin approximation. For $k=0$, the order remains the same.