Journal of Computational Physics
A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation
SIAM Journal on Numerical Analysis
Parallel, adaptive finite element methods for conservation laws
Proceedings of the third ARO workshop on Adaptive methods for partial differential equations
A parallel hp-adaptive discontinuous Galerkin method for hyperbolic conservation laws
Applied Numerical Mathematics - Special issue on adaptive mesh refinement methods for CFD applications
Parallel adaptive hp-refinement techniques for conservation laws
Applied Numerical Mathematics - Special issue on adaptive mesh refinement methods for CFD applications
Journal of Computational Physics
Journal of Parallel and Distributed Computing - Special issue on dynamic load balancing
The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems
SIAM Journal on Numerical Analysis
A Local Discontinuous Galerkin Method for KdV Type Equations
SIAM Journal on Numerical Analysis
Superconvergence of the Local Discontinuous Galerkin Method for Elliptic Problems on Cartesian Grids
SIAM Journal on Numerical Analysis
An A Priori Error Analysis of the Local Discontinuous Galerkin Method for Elliptic Problems
SIAM Journal on Numerical Analysis
Journal of Scientific Computing
Journal of Scientific Computing
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
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
A Superconvergent Local Discontinuous Galerkin Method for Elliptic Problems
Journal of Scientific Computing
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In this paper we investigate the superconvergence properties of the local discontinuous Galerkin (LDG) method applied to transient convection-diffusion problems in one space dimension. We show that the leading terms of the local discretization errors for the p-degree LDG solution and its spatial derivative are proportional to (p+1)-degree right and left Radau polynomials, respectively. Thus, the discretization errors for the p-degree LDG solution and its spatial derivative are O(h^p^+^2) superconvergent at the roots of (p+1)-degree right and left Radau polynomials, respectively. The superconvergence results are used to construct asymptotically correct a posteriori error estimates. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary conditions on each element. Numerical computations suggest that these a posteriori LDG error estimates for the solution and its spatial derivative at a fixed time t converge to the true errors at O(h^p^+^3) and O(h^p^+^2) rates, respectively. We also show that the global effectivity indices for the solution and its derivative in the L^2-norm converge to unity at O(h^2) and O(h) rates, respectively. Finally, we show that the LDG method combined with the a posteriori error estimation procedure yields both accurate error estimates and O(h^p^+^2) superconvergent solutions. Our proofs are valid for arbitrary regular meshes and for P^p polynomials with p=1, and for periodic, Dirichlet, and mixed Dirichlet-Neumann boundary conditions. Several numerical simulations are performed to validate the theory.