Discontinuous Galerkin Method： Chi－Wang Shu

is used for conservation laws and when the alternating flux is used for convection-diffusion

equations. The order of superconvergence for both cases is proved to be k+

P

meshes and for piecewise P

in [8, 9] in which the proof based on Fourier analysis was given only for uniform meshes and

FG02-08ER25863.

In this paper, we consider one-dimensional linear hyperbolic conservation laws

and convection-diffusion equations

discontinuous piecewise polynomial space for the numerical solution and the test functions.

It is originally devised to solve hyperbolic conservation laws containing only first order spatial

derivatives, e.g. [14, 13, 12, 11, 15, 17]. It has the advantage of flexibility for arbitrarily

unstructured meshes, with a compact stencil, and with the ability to easily accommodate

arbitrary h-p adaptivity. The DG method was later generalized to the LDG method by

Cockburn and Shu to solve the convection-diffusion equation [16]. Their work was motivated

by the successful numerical experiments of Bassi and Rebay [5] for the compressible Navier-

Stokes equations.

convergent at Radau points. In [8], we proved superconvergence of the DG solution towards

two dimensions. For convection-diffusion equations, in [7], Celiker and Cockburn studied the

steady state solution of (1.2), and proved that for a large class of DG methods, the numerical

fluxes (traces) are superconvergent, also see [6] for related discussions on elliptic problems.

superconvergence property of the LDG scheme for convection-diffusion equations. We proved

the superconvergence result for the heat equation in the case of piecewise linear solutions

on uniform meshes, and gave numerical tests to demonstrate the validity of the result for

higher order schemes and nonlinear equations.

space and uniform meshes. In this paper, we use a different framework to prove the super-

convergence results and do not rely on Fourier analysis. The proof now works for arbitrary

non-uniform regular meshes and schemes of any order.

(1.2), the same superconvergence results can be easily proved for one-dimensional linear

gence. Finally, conclusions and plans for future work are provided in Section 4. The proofs

for some of the technical lemmas are collected in the Appendix.

In this section, we consider, without loss of generality, the linear conservation law (1.1) with

c = 1:

the interval I = [a, b], first we divide it into N cells as follows

We denote

as the cells and cell centers respectively. h

P

flux will become: find u

Notice that this special projection is used in the error estimates of the DG methods to derive

optimal L

the numerical solution is closer to this special projection of the exact solution than to the

and ¯e = P

exact solution.

2.1 that they are related to the L

Theorem 2.2. Let u be the exact solution of the equation (2.1). If k ≥ 1, define u

the DG solution of (2.4) with the initial condition u

and

where C