Multiblock Modeling of Flow in Porous Media and Applications

We investigate modeling flow in porous media in multiblock domain. Mixed finite element methods are used for subdomain discretizations. Physically meaningful boundary conditions are imposed on the non-matching interfaces via mortar finite element spaces. We investigate the pollution effect of nonmatching grids error on the numerical solution away from interfaces. We prove that most of the error in the velocity occurs along the interfaces, and that high accuracy is preserved in the interior of the subdomains. In case of discontinuous coefficients, the pollution from the singularity affects the accuracy in the whole domain. We investigate the upscaling error resulting when fine resolution data is approximated on a very coarse scale. Extending work of Wheeler and Yotov, we incorporate this upscaling error in an a posteriori error estimator for the pressure, velocity and mortar pressure. We employ a non-overlapping domain decomposition method reducing the global system to one that is solved iteratively via a preconditioned conjugate gradient method. This approach is suitable for parallel implementation. The balancing domain decomposition method for mixed finite elements following Cowsar, Mandel, and Wheeler is extended to the case of mortar mixed finite elements on non-matching multiblock grids. The algorithm involves solution of a mortar interface problem with one local Dirichlet solve and one local Neumann solve on each iteration. A coarse solve is used to guarantee consistency and to provide global exchange of information. Quasi-optimal condition number bounds independent of the jump in coefficients are derived. iii We finally consider multiscale mortar mixed finite element discretizations for single and two phase flows. We show optimal convergence and some superconvergence in the fine scale for the solution and its flux. We also derive efficient and reliable a posteriori error estimators suitable for adaptive mesh refinement. We have incorporated the above methods into a parallel multiblock simulator on un-structured prismatic meshes employing a non-overlapping domain decomposition algorithm and mortar spaces. Numerical experiments are presented confirming all theoretical results. iv To my husband Dimitar and my daughter Kalina v Acknowledgments