Nečas Center for Mathematical Modeling An Optimal L∞(L2) - Error Estimate for the DG Approximation of a Nonlinear Nonstationary Convection-Diffusion Problem on Nonconforming Meshes

This 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 Dirichlet problem. General nonconforming simplicial meshes are considered and the SIPG scheme is used. Under the assumption that the exact solution is sufficiently regular an L∞(L2)-optimal error estimate is derived. The theoretical results are illustrated by numerical experiments. 1991 Mathematics Subject Classification. 65M12, 65M15, 65M60. . INTRODUCTION During the last decade, the discontinuous Galerkin finite element method (DGFEM) for the numerical solution of partial differential equations has become rather popular because of several attractive properties. It represents a natural generalization of the finite volume and finite element techniques. Similarly to the finite volume method, the DGFEM uses discontinuous approximations, it does not require continuity on common boundaries of two neighbouring elements, and the fluxes on these boundaries are evaluated with the aid of a numerical flux. On the other hand, the application of higher degree polynomials, which is common with the finite element method, produces high order of accuracy in regions where the solution is smooth. Concerning theoretical aspects of the DGFEM, let us mention, e. g. the important survey papers [9] and [2] and the works [1], [3], [4], [6], [16], [17], [18], [19], [20], [22], [24], [25], [26]. For a survey of various discontinuous Galerkin techniques, see, e. g. [8], [10]. It occurs that the DGFEM is very suitable for the numerical solution of nonlinear convection-diffusion problems, conservation law equations and compressible flow. The papers [12], [13] and [15] are concerned with the analysis of the DGFEM for the solution of nonstationary nonlinear convection-diffusion problems. The approximation of nonlinear convective fluxes was carried out with the aid of a numerical flux. In the discretization of the diffusion term both the