Discontinuous Galerkin method for nonlinear convection-diffusion problems with mixed Dirichlet-Neumann boundary conditions

The paper is devoted to the analysis of the discontinuous Galerkin finite element method (DGFEM) applied to the space semidiscretization of a nonlinear nonstationary convection-diffusion problem with mixed Dirichlet-Neumann boundary conditions. General nonconforming meshes are used and the NIPG, IIPG and SIPG versions of the discretization of diffusion terms are considered. The main attention is paid to the impact of the Neumann boundary condition prescribed on a part of the boundary on the truncation error in the approximation of the nonlinear convective terms. The estimate of this error allows to analyse the error estimate of the method. The results obtained represent the completion and extension of the analysis from V. Dolejší, M. Feistauer, Numer. Funct. Anal. Optim. 26 (2005), 349–383, where the truncation error in the approximation of the nonlinear convection terms was proved only in the case when the Dirichlet boundary condition on the whole boundary of the computational domain was considered.