Discontinuous Galerkin Methods for Advection-diffusion-reaction Problems

We apply the weighted-residual approach recently introduced in [7] to derive dis-continuous Galerkin formulations for advection-diffusion-reaction problems. We devise the basic ingredients to ensure stability and optimal error estimates in suitable norms, and propose two new methods. 1. Introduction. In recent years Discontinuous Galerkin methods have become increasingly popular, and they have been used and analyzed for various kinds of applications: see, e.g., [2] for second order elliptic problems, [4]-[3] for Reissner-Mindlin plates and, for advection-diffusion problems, Most DG-methods for advection-diffusion or hyperbolic problems are constructed by specifying the numerical fluxes at the inter-elements and, as far as we know, the ad-vection field is always assumed to be either constant or divergence free. In the present paper we follow a different path. From the one hand, we derive DG-formulations by applying the so called " weighted-residual " approach of [7]. In this approach a DG-method is written first in strong form, as a system of equations including the original PDE equation inside each element plus the necessary continuity conditions at interfaces. The variational form is then obtained by combining all these equations. In this way, the DG-method establishes a linear relationship between the residual inside each element and the jumps across inter-element boundaries. Such a linear relation permits to recover DG-methods proposed earlier in literature, and at the same time provides a framework for devising new DG-methods with the desired stability and consistency properties. As we shall show, this is possible, since stability and consistency can be ensured through a proper selection of the weights in the linear relationship, which in turn determines the DG-method. On the other hand, and this is, in our opinion, the novelty of the present paper, we deal with a variable advection field which is not divergence-free. This, together with the presence of a variable reaction, makes the analysis more complicated than usually, surely more complicated than one could expect at first sight. To ease the presentation we apply the " weighted-residual " approach to derive two DG-methods proposed in literature: the method introduced in [17], and that proposed in [18] and further analyzed in [9]. The former uses the non-symmetric NIPG method for the diffusion terms and upwind for the convective part of the flux. In the latter the diffusion terms are treated with three different DG-methods, and the whole physical flux is upwinded. This makes the approach well suited for strongly advection dominated …