Title:
    Mixed Finite Element - Finite Volume Methods for Two Phase
    Flow Problems in Porous Media

Abstract:
    As a model problem for the miscible and immiscible two
    phase flow we consider the following system of partial
    differential equations:
    \begin{eqnarray}
        \mbox{div}\ u(x,t) &=& 0 \nonumber \\
        u(x,t) &=& -a(c,x) \ (\nabla  p(x,t) + \gamma(c, x)) \nonumber \\
        \partial_t c(x,t) + \mbox{div}\ (u(x,t) c(x,t))
                &-& \mbox{div}\ (\epsilon \ \nabla c(x,t)) = f(x,t), \nonumber
    \end{eqnarray}
    with $(x,t) \in \Omega \times (0,T)$.
    Here $u$ denotes the Darcy velocity,
    $p$ the pressure and $c$ the concentration of one phase of the fluid.
    Considering density driven flow or immiscible flow of water and oil
    in a reservoir the convection of the concentration is dominant to the
    diffusion. Thus we have to look at this system of partial differential
    equations as a singular perturbed problem in $\epsilon$.
    For small diffusions ($0 < \epsilon <\!< 1$) standard Galerkin Finite
    Element approximations do not produce stable solutions.
    Therefore we propose a combined Mixed Finite Element - Finite
    Volume discretisation,  specifically to handle this convection dominated
    diffusion problem.\\
    Taking into account the dependence on the diffusion parameter
    $\epsilon$ we prove a convergence result for this first order
    scheme on triangular meshes.