Analysis of variable-degree HDG methods for convection-diffusion equations. Part I: General nonconforming meshes

In this paper the first error analyses of hybridizable discontinuous Galerkin (HDG) methods for convection-diffusion equations for variable-degree approximations and nonconforming meshes are presented. The analysis technique is an extension of the projection-based approach recently used to analyse the HDG method for the purely diffusive case. In particular, for approximations of degree k on all elements and conforming meshes, we show that the order of convergence of the error in the diffusive flux is k + 1 and that of a projection of the error in the scalar unknown is 1 for k = 0 and k + 2 for k > 0. When nonconforming meshes are used our estimates do not rule out a degradation of 1/2 in the order of convergence in the diffusive flux and a loss of 1 in the order of convergence of the projection of the error in the scalar variable. However, they do guarantee the optimal convergence of order k + 1 of the scalar variable.