On some aspects of the discontinuous Galerkin finite element method for conservation laws

The paper is concerned with the numerical solution of nonlinear conservation laws and nonlinear convection–diffusion problems. We discuss two versions of this method: (a) Finite volume discontinuous Galerkin (FVDG) method, which is a generalization of the combined finite volume–finite element (FV–FE) method. Its advantage is the use of only one mesh (in contrast to the combined FV–FE schemes). However, it is of the first order only. (b) Further, the pure DGFE method of higher order is considered. In this case, a new limiting is developed to avoid spurious oscillations in the vicinity of shocks.