We show that the approximation given by the original discontinuous Galerkin method for the transport-reaction equation in d space dimensions is optimal provided the meshes are suitably chosen: the L2-norm of the error is of order k + 1 when the method uses polynomials of degree k. These meshes are not necessarily conforming and do not satisfy any uniformity condition; they are only required to ...

Classical nite element methods rely on tessellations composed of straight-edged elements mapped linearly from a reference element, on domains which physical boundaries are indi erently straight or curved. This approximation represents serious hindrance for high-order methods, since they limit the precision of the spatial discretization to second order. Thus, exploiting an enhanced representatio...

The local discontinuous Galerkin (LDG) viscous flux formulation was originally developed by Cockburn and Shu for the discontinuous Galerkin setting and later extended to the spectral volume setting by Wang and his collaborators. Unlike the penalty formulations like the interior penalty and the BR2 schemes, the LDG formulation requires no length based penalizing terms and is compact. However, co...

We analyze the classical discontinuous Galerkin method for a general parabolic equation. Symmetric error estimates for schemes of arbitrary order are presented. The ideas we develop allow us to relax many assumptions freqently required in previous work. For example, we allow different discrete spaces to be used at each time step and do not require the spatial operator to be self adjoint or inde...

We report on recent efforts towards the development of a high order, non-conforming, discontinuous Galerkin method for the solution of the system of frequency domain Maxwell’s equations in heterogeneous propagation media. This method is an extension of the low order one which was proposed in [1].

In this paper we present the analysis for the Runge-Kutta discontinuous Galerkin (RKDG) method to solve scalar conservation laws, where the time discretization is the third order explicit total variation diminishing Runge–Kutta (TVDRK3) method. We use an energy technique to present the L-norm stability for scalar linear conservation laws, and obtain a priori error estimates for smooth solutions...

We present a space-time finite element formulation of the Navier-Stokes equations which is stabilized via the Galerkin/least-squares approach. The variational equation is based on the time discontinuous Galerkin method and is written in terms of the physical entropy variables over the moving and deforming space-time slabs. The formulation thus becomes analogous to the classical arbitrary Lagran...

In this paper we present an error estimate for the explicit Runge-Kutta discontinuous Galerkin method to solve linear hyperbolic equation in one dimension with discontinuous but piecewise smooth initial data. The discontinuous finite element space is made up of piecewise polynomials of arbitrary degree, and time is advanced by the third order explicit total variation diminishing Runge-Kutta met...

We study time step restrictions due to linear stability constraints of Runge-Kutta Discontinuous Galerkin methods on triangular grids. The scalar advection equation is discretized in space by the Discontinuous Galerkin method with either the Lax-Friedrichs flux or the upwind flux, and integrated in time with various Runge-Kutta schemes designed for linear wave propagation problems or non-linear...

Abstract. We propose a general method for the design of Discontinuous Galerkin Methods for non stationary linear equations. The method is based on a particular splitting of the bilinear forms that appear in the weak Discontinuous Galerkin Method. We prove that an appropriate time splitting gives a stable scheme whatever the order of the polynomial approximation . Various problems can be address...