An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation
Mathematics of Computation
High-order finite element methods for singularly perturbed elliptic and parabolic problems
SIAM Journal on Applied Mathematics
The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems
SIAM Journal on Numerical Analysis
The Convergence Rate of Finite Difference Schemes in the Presence of Shocks
SIAM Journal on Numerical Analysis
Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
Journal of Scientific Computing
Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
Journal of Scientific Computing
Journal of Scientific Computing
Locally divergence-free discontinuous Galerkin methods for the Maxwell equations
Journal of Computational Physics
Journal of Computational Physics
Journal of Scientific Computing
Local derivative post-processing for the discontinuous Galerkin method
Journal of Computational Physics
SIAM Journal on Scientific Computing
SIAM Journal on Numerical Analysis
SIAM Journal on Scientific Computing
On the Negative-Order Norm Accuracy of a Local-Structure-Preserving LDG Method
Journal of Scientific Computing
Journal of Scientific Computing
Negative-Order Norm Estimates for Nonlinear Hyperbolic Conservation Laws
Journal of Scientific Computing
Journal of Computational and Applied Mathematics
A scalable, efficient scheme for evaluation of stencil computations over unstructured meshes
SC '13 Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis
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We consider the enhancement of accuracy, by means of a simple post-processing technique, for finite element approximations to transient hyperbolic equations. The post-processing is a convolution with a kernel whose support has measure of order one in the case of arbitrary unstructured meshes; if the mesh is locally translation invariant, the support of the kernel is a cube whose edges are of size of the order of Δx only. For example, when polynomials of degree k are used in the discontinuous Galerkin (DG) method, and the exact solution is globally smooth, the DG method is of order k+1/2 in the L2-norm, whereas the post-processed approximation is of order 2k + 1; if the exact solution is in L2 only, in which case no order of convergence is available for the DG method, the post-processed approximation converges with order k + 1/2 in L2(Ω0), where Ω0 is a subdomain over which the exact solution is smooth. Numerical results displaying the sharpness of the estimates are presented.