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

In this paper we investigate the superconvergence properties of the discontinuous Galerkin method applied to scalar first-order hyperbolic partial differential equations on triangular meshes. We show that the discontinuous finite element solution is O(h ^p+2) superconvergent at the Legendre points on the outflow edge for triangles having one outflow edge. For triangles having two outflow edges the finite element error is O(h ^p+2) superconvergent at the end points of the inflow edge. Several numerical simulations are performed to validate the theory. In Part II of this work we explicitly write down a basis for the leading term of the error and construct asymptotically correct a posteriori error estimates by solving local hyperbolic problems with no boundary conditions on more general meshes.