Space-time discontinuos Galerkin method for solving nonstationary convection-diffusion-reaction problems

The paper presents the theory of the discontinuous Galerkin finite element method for the space-time discretization of a linear nonstationary convection- diffusion-reaction initial-boundary value problem. The discontinuous Galerkin method is applied separately in space and time using, in general, different nonconforming space grids on different time levels and different polynomial degrees p and q in space and time discretization, respectively. In the space discretization the nonsymmetric interior and boundary penalty approximation of diffusion terms is used. The paper is concerned with the proof of error estimates in "L-2(L-2)"- and "root epsilon L-2 (H-1)"-norms, where epsilon >= 0 is the diffusion coefficient. Using special interpolation theorems for the space as well as time discretization, we find that under some assumptions on the shape regularity of the meshes and a certain regularity of the exact solution, the errors are of order O(h(p) + r(q)). The estimates hold true even in the hyperbolic case when epsilon = 0.