A Hybridizable Discontinuous Galerkin Method for Steady-State Convection-Diffusion-Reaction Problems

Abstract

In this article, we propose a novel discontinuous Galerkin method for convectiondiffusion-reaction problems, characterized by three main properties. The first is that the method is hybridizable; this renders it efficiently implementable and competitive with the main existing methods for these problems. The second is that, when the method uses polynomial approximations of the same degree for both the total flux and the scalar variable, optimal convergence properties are obtained for both variables; this is in sharp contrast with all other discontinuous methods for this problem. The third is that the method exhibits superconvergence properties of the approximation to the scalar variable; this allows us to postprocess the approximation in an element-by-element fashion to obtain another approximation to the scalar variable which converges faster than the original one. In this paper, we focus on the efficient implementation of the method and on the validation of its computational performance. With this aim, extensive numerical tests are devoted to explore the convergence properties of the novel scheme, to compare it with other methods in the diffusiondominated regime, and to display its stability and accuracy in the convection-dominated case.

To CG or to HDG: A Comparative Study

“…In the next section, we show how to use our elemental local solvers given by (21) and (22) to obtain a matrix equation for λ only.…”

“…Therefore we consider a value of τ = 1 with the case when τ = 1000 (which is close to the CG solution) for the Poisson equation when λ = 0 with the exact solution given by (35). We recall that a comparison of the τ = 1 HDG with CG method was also considered in [21] for up to third-order polynomial expansions. In Fig.…”

“…To the best knowledge of the authors, the only comparison of the CG and HDG methods was carried out in [21], for a convection-diffusion-reaction model problem in a diffusiondominated regime. This paper however only considers up to third-order polynomial expansions and considers counting non-zero matrix entries as a reflection of computation time rather than any direct timing comparisons.…”

Optimization of a regularized distortion measure to generate curved high‐order unstructured tetrahedral meshes

“…Then, we use this mesh to obtain an approximation of the steady state solution of the Euler equations. This approximation is obtained with a 3D and parallel solver [60] using the hybridized discontinuous Galerkin (HDG) method [61,62,63,64]. (21) points.…”

Central local discontinuous galerkin methods on overlapping cells for diffusion equations

“…attracted a lot of attention in the literature, for example we mention the recovery type DG method by van Leer and Nomura [8] (see also [9]), and the hybridizable discontinuous Galerkin (HDG) method of Cockburn et al [5].…”