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

This paper is dedicated to Chi-Wang Shu in the occasion of his 65th birthdayIn 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 0, 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.