Turbo Post-processing for Discontinuous Galerkin Methods: One-Dimensional Linear Transport

We discover a new type of post-processing for the discretization in space of the one-dimensional transport equation, with periodic boundary conditions, by the discontinuous Galerkin method with polynomials of degree k≥0. We prove that the post-processing is locally conservative and that it provides an approximation of order 2k+1, even for unstructured meshes. The post-processing is computed only when the approximation is needed. We refer to it as the Turbo Post-Processing because its computation is extremely fast. Unlike the well-known post-processing based on convolutions, see Bramble and Schatz (Math Comput 31:94–111, 1977), our post-processing does not require locally uniform meshes, and is devised by using the idea of transforming stabilizations into spaces introduced in Cockburn (Jpn J Ind Appl Math 42:1637–1676, 2023), in the framework of second-order elliptic equations. We carry out numerical experiments which validate the predicted theoretical orders of convergence as the meshes are refined. We end by discussing several forthcoming extensions.