A Superconvergent Discontinuous Galerkin Method for Hyperbolic Problems on Tetrahedral Meshes

In this manuscript we present a superconvergent discontinuous Galerkin method equipped with an element residual error estimator applied to scalar first-order hyperbolic problems using tetrahedral meshes. We present a local error analysis to derive a discontinuous Galerkin orthogonality condition for the leading term of the discretization error and establish new superconvergence points, lines and surfaces. We also derive new basis functions spanning the error and propose an implicit error estimation procedure by solving a local problem on each tetrahedron. The DG method combined with the a posteriori error estimation procedure yields both accurate error estimates and O(hp+2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(h^{p+2})$$\end{document} superconvergent solutions. Computations validate our theory.