2025/09/26 by Bernardo Cockburn, Cockburn, Bernardo
Engineering · Mathematics · #35L65 #65M60 #Advanced Numerical Methods in Computational Mathematics #Differential Equations and Numerical Methods #F.2.1 #FOS: Mathematics #Fractional Differential Equations Solutions #Numerical Analysis (math.NA) #Primary 65N30
paper · pdf · doi:10.48550/arxiv.2509.22587
openalex publication_date 2025/09/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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≥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.