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 $$k+1$$k+1 and $$k\ge 0$$k?0 to approximate the scalar variable and its gradient, respectively. In contrast, we only use polynomials of degree $$k$$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 $$L^{2}$$L2-norm of the error of the scalar variable converges with the order of $$k+2$$k+2 while that of its gradient converges with order $$k+1$$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 Poincare 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.