An HDG Method for Convection Diffusion Equation

Abstract

We present a new hybridizable discontinuous Galerkin (HDG) method for the convection diffusion problem on general polyhedral meshes. This new HDG method is a generalization of HDG methods for linear elasticity introduced in Qiu and Shi (2013) to problems with convection term. For arbitrary polyhedral elements, we use polynomials of degree k + 1 and k ≥ 0 to approximate the scalar variable and its gradient, respectively. In contrast, we only use polynomials of degree k to approximate the numerical trace of the scalar variable on the faces which allows for a very efficient implementation of the method, since the numerical trace of the scalar variable is the only globally coupled unknown. The global L 2 -norm of the error of the scalar variable converges with the order of k + 2 while that of its gradient converges with order k + 1. From the point of view of degrees of freedom of the globally coupled unknown: numerical trace, this method achieves superconvergence for the scalar variable without postprocessing. A key inequality relevant to the discrete Poincaré inequality is a novel theoretical contribution. This inequality is useful to deal with convection term in this paper and is essential to error analysis of HDG methods for the Navier-Stokes equations and other nonlinear problems.

A Superconvergent Ensemble HDG Method for Parameterized Convection Diffusion Equations

“…We emphasize that the HDG method in this work is considered to be a superconvergent method. Specifically, if polynomials of degree k ≥ 0 are used for the numerical traces (global system), then we can obtain k + 2 order for the scalar variables; see, e.g., [22][23][24]. Hence, from the viewpoint of globally coupled degrees of freedom, this method achieves superconvergence for the scalar variable.…”

“…In this paper, we first restore the superconvergence for k = 0 by modifying the stabilization function in [23]. Next, we show that the new ensemble HDG method can obtain a L ∞ (0, T ; L 2 (Ω)) superconvergent rate for all k ≥ 0 on a general polyhedron mesh and without assume the coefficients are independent of time.…”

“…This method was proposed by [19] and later analyzed by [21] for a single steady elliptic PDE, they obtained a superconvergent rate for the scalar variable for all k ≥ 0. This HDG method has been extended to study the PDEs with a convection term by [23,24]. However, the superconvergent rate was lost when k = 0.…”

A superconvergent HDG method for the incompressible Navier–Stokes equations on general polyhedral meshes

“…First, we would like to recall an important inequality which was introduced in [26]. Here we write it in a slightly general way.…”

“…For the proof of the above result, we refer the Lemma 3.2 in [26]. In addition, we also need the following basic inequalities:…”