Theory of the Space-Time Discontinuous Galerkin Method for Nonstationary Parabolic Problems with Nonlinear Convection and Diffusion

Abstract

The paper presents the theory of the space-time discontinuous Galerkin finite element method for the discretization of a nonstationary convection-diffusion initial-boundary value problem with nonlinear convection and nonlinear diffusion. The discontinuous Galerkin method is applied separately in space and time using, in general, different space grids on different time levels and different polynomial degrees p and q in space and time discretization. In the space discretization the nonsymmetric, symmetric, and incomplete interior and boundary penalty approximations of diffusion terms are used. The paper is concerned with the analysis of error estimates in the “$L^2(L^2)$”- and “DG”-norm formed by the “$L^2(H^1) $”-seminorm and penalty terms. An important ingredient used in the derivation of the error estimate is the concept of the discrete characteristic function and its properties. In the “DG”-norm the error estimates are optimal with respect to the size of the space grid. They are optimal with respect to the time step if the Dirichlet boundary condition behaves in time as a polynomial of degree $\leq q$. In a general case, the derived estimates become suboptimal in time.