A mixed finite volume element method for accurate computation of fluid velocities in porous media

A key ingredient in the simulation of flow in porous media is the accurate determination of the velocities that drive the flow. Large-scale irregularities of the geology, such as faults, fractures, and layers, suggest the use of irregular grids in the simulation. In this study, the approach to this problem was to apply the finite volume element methodology, developed by McCormick, in conjunction with mixed methods, developed by Raviart and Thomas. The resulting mixed finite volume element discretization scheme developed here can be applied in a clear and straightforward way to irregular grids and is appealing because of its local conservation properties and its direct and accurate representation of physical intercell flux terms. Several multilevel algorithms are developed that provide efficient methods for solving the set of equations that this discretization produces. This thesis includes numerical results from a variety of test problems, from Poisson's equation to problems with anisotropic, discontinuous, and tensor diffusion coefficients. These results show that this approach has the potential to generate accurate approximate fluid velocities and that the multilevel methods can provide fast solvers.