Optimal Convergence of the Original DG Method for the Transport-Reaction Equation on Special Meshes

Abstract

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 L 2-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 be made of simplexes each of which has a unique outflow face. We also find a new, element-by-element postprocessing of the derivative in the direction of the flow which superconverges with order k + 1.

On the order of convergence of the discontinuous Galerkin method for hyperbolic equations

“…The triangulation employed in [4] was uniform, except for a set of characteristic mesh lines spaced at O(h σ ) intervals, σ ∈ [0, 1]. The optimal O(h 2 ) convergence rate for linears was observed for σ = 1, in a similar vein to the theoretical result in [1]. However, taking σ = .75 as h → 0 produced order h 1.5 convergence, as predicted by (3).…”

“…With these assumptions, the error generated in each layer of triangles is damped exponentially as u h evolves through subsequent layers, leading to the optimal rate. Recently, Cockburn, et al [1] derived an optimal order estimate, requiring only u ∈ H n+1 (Ω), for a special mesh in R d consisting of simplices having one outflow face. For d = 2, such a mesh can be constructed by arranging triangles along characteristic mesh lines with O(h) spacing.…”

“…in Ω, u = g, on Γ in (Ω), (1) where Ω is a 2-dimensional polygonal domain and α (a vector) and β are constants.…”

A Space-Time Discontinuous Galerkin Method for First Order Hyperbolic Systems

“…However, for a scalar hyperbolic equation, this order of convergence can be further improved if some special structured meshes are used, see [4,12].…”

Optimal convergence and a posteriori error analysis of the original DG method for advection-reaction equations

“…Obviously, condition (1.3) is less significant in the practical case, and the regularity required for the exact solution in (1.2) is not optimal. The first optimal convergence is obtained by Cockburn et al in [7] under the assumptions that β is a constant vector and the triangulation T h satisfies the socalled flow condition: (1.4) Each simplex K has a unique outflow face e + K with respect to β and there are no hanging nodes on each interior outflow face e…”