Spectral methods on triangles and other domains
Journal of Scientific Computing
The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Journal of Scientific Computing
Journal of Scientific Computing
Superconvergence of Discontinuous Galerkin Methods for Convection-Diffusion Problems
Journal of Scientific Computing
Asymptotically exact a posteriori error estimates for a one-dimensional linear hyperbolic problem
Applied Numerical Mathematics
Negative-Order Norm Estimates for Nonlinear Hyperbolic Conservation Laws
Journal of Scientific Computing
Journal of Computational and Applied Mathematics
A Superconvergent Discontinuous Galerkin Method for Hyperbolic Problems on Tetrahedral Meshes
Journal of Scientific Computing
Asymptotically exact discontinuous Galerkin error estimates for linear symmetric hyperbolic systems
Applied Numerical Mathematics
Computers & Mathematics with Applications
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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.