Characteristic mixed discontinuous finite element methods for advection-dominated diffusion problems Original Research Article

In this paper three characteristic mixed discontinuous finite element methods are introduced for time dependent advection-dominated diffusion problems. Namely, the diffusion term in these problems is discretized using mixed discontinuous finite elements, and the temporal differentiation and advection terms are treated by a characteristic tracking scheme. The first method is based on the standard modified method of characteristics. It is simple to set up and analyze, but fails to preserve an integral identity satisfied by the solution of the differential problems. The second method is formulated using the modified method of characteristics with adjusted advection and preserves the integral identity globally. The third method is defined in terms of a local Eulerian–Lagrangian technique and preserves the identity locally. These three methods not only preserve the conceptual and computational merits of both characteristics-based procedures and discontinuous finite element schemes, they also possess new features such as they are more stable, are more accurate, and can handle the case where the diffusion coefficient is zero. Stability and convergence properties are studied for all three methods; unconditionally stable results and sharp error estimates are established. Their relationships to standard characteristics-based methods such as MMOC, MMOCAA, ELLAM, and CMFEM are described in detail. Numerical results are presented.