The Discontinuous Galerkin Method for Two-Dimensional Hyperbolic Problems. Part I: Superconvergence Error Analysis

A stabilization for three‐dimensional discontinuous Galerkin discretizations applied to nonhydrostatic atmospheric simulations

“…When the resolution is sufficient to attain the convergence zone, the model exhibits a convergence rate of about P + 1 for a polynomial order P (Figure ). Some convergence rates are even higher than P + 1, which is probably related to the superconvergence properties of the DG method .…”

“…As the model solution is close to the analytical solution, the difference between them is only visible for the largest times (lower panels). are even higher than P C 1, which is probably related to the superconvergence properties of the DG method [56]. Based on those convergence results, the spatial discretization seems optimal.…”

A superconvergent local discontinuous Galerkin method for the second-order wave equation on Cartesian grids

“…For an introduction to the subject of a posteriori error estimation see the monograph of Ainsworth and Oden [19]. Superconvergence properties for finite element and DG methods have been studied in [3,5,20,2] for ordinary differential equations, [21,22,3,23] for hyperbolic problems and [24][25][26]23,[27][28][29][30][31] for diffusion and convection-diffusion problems. Several a posteriori DG error estimates are known for hyperbolic [32][33][34] and diffusive [35,36] problems.…”

“…They identified a special numerical flux for which the L 2 -norms of the gradient and the potential are of orders p+1/2 and p+1, respectively, when tensor product polynomials of degree at most p are used. Adjerid and Baccouch [24,25,28] investigated the superconvergence properties of DG solutions of a scalar steady first-order hyperbolic model problem on structured and unstructured triangular meshes. They presented a detailed discussion on the superconvergence properties versus the choice of finite element polynomial spaces.…”

Discontinuous Galerkin error estimation for linear symmetrizable hyperbolic systems

“…They computed DG error estimates by solving local problems involving numerical fluxes, thus requiring information from neighboring inflow elements. Adjerid and Baccouch [2,4] investigated DG methods on structured and unstructured triangular meshes with several finite element spaces to discover new superconvergence properties and compute efficient and accurate error estimates. Cheng and Shu [12,13,14] investigated the superconvergence of discontinuous solutions for hyperbolic and convection-diffusion problems.…”