Proceedings of ALGORITMY 2009 pp. 294–303 DISCONTINUOUS GALERKIN METHOD FOR NONSTATIONARY NONLINEAR CONVECTION–DIFFUSION PROBLEMS: A PRIORI ERROR ESTIMATES * JI ˇ R ´ I HOZMAN † Abstract. We deal with a numerical solution of a scalar nonstationary convection-diffusion equation with nonlinear convective as well as diffusive terms which represents a model problem for the solution of the system of the compressible Navier-Stokes equations describing a motion of viscous compressible fluids. We present a discretization of this model equation by the interior penalty discontinuous Galerkin methods. Moreover, under some assumptions on the nonlinear terms, domain partitions and the regularity of the exact solution, we introduce a priori error estimates in the L ∞ (0,T ; L 2 (Ω))-norm and in the L 2 (0,T ; H 1 (Ω))-semi-norm. A sketch of the proof and numerical verifications are presented. Key words. discontinuous Galerkin method, convection-diffusion problem, a priori error esti- mates AMS subject classifications. 65M60, 65M15, 65M12, 65M20 1. Introduction. Our goal is to develop a sufficiently robust, accurate and ef- ficient numerical method for the solution of the system of the compressible Navier- Stokes equations describing a motion of viscous compressible fluids. Due to the lack of the theory concerning with an existence of the solution of the Navier-Stokes equa- tions we consider the model problem represented by a nonstationary two-dimensional convection–diffusion equation with nonlinear convection as well as diffusion. Among a wide class of numerical methods, the discontinuous Galerkin finite element method (DGFEM) seems to be a promising technique for the solution of convection-diffusion problems. DGFEM is based on a piecewise polynomial but dis- continuous approximation, for a survey, see, e.g., [4], [5]. Within this paper we deal with the space semidiscretization of the model problem with the aid three variants of DGFEM, namely nonsymmetric (NIPG), symmetric (SIPG) and incomplete interior penalty Galerkin (IIPG) techniques, see [1]. This article represents a generalization of research papers [7], [8], [9], [10], where the linear diffusion term was considered. Moreover, let us cite works [6], [11], [12], where simpler forms of nonlinear diffusion were analysed. 2. Problem formulation. We consider the following unsteady nonlinear con- vection-diffusion problem: Let Ω ⊂ IR 2 be a bounded polygonal domain, T> 0, we seek a function u : Q T =Ω × (0,T ) → IR such that ∂u ∂t + 2  s=1 ∂f s (u) ∂x s = div(IK(u) ∇u)+ g in Q T , (1) u   ∂Ω×(0,T ) = u D , (2) * This work is a part of the research project MSM 0021620839 financed by the Ministry of Educa- tion of the Czech Republic and it was partly supported by the Grant No. 316/2006/B-MAT/MFF of the Grant Agency of the Charles University Prague. † Department of Numerical Mathematics, Faculty of Mathematics and Physics, Charles University Prague, Sokolovsk´ a 83, Prague, 186 75, Czech Republic (jhozmi@volny.cz). 294