Preprint C67/2008
DOMAIN DECOMPOSITION ANALYSIS FOR HETEROGENEOUS DARCY'S FLOW
Juan Carlos Galvis Arrieta
Keywords: finite element | porous media flow | multiphysics | inf-sup | error estimates | non-matching grids | Stokes-Darcy | domain decomposition preconditioners | discontinuous coefficients | discontinuous Galerkin | white noise analysis | Chaos expansion
This thesis focuses on finite element and domain decomposition applications to three mathematical models related to fluid flow in porous media. The models have application in a variety of fields including areas such as petroleum engineering, environmental sciences, hydrology and biology, among others. The first model considered is the Darcy-Stokes coupling. We study the well posedness of the continuous model. We introduce a discretization, obtain the well posedness of the discrete model and derive a priori error estimates. We also design and analyze two domain decomposition preconditioners. The second model is the pressure equation with discontinuous coefficients. Here we design and analyze several domain decomposition preconditioners for the resulting linear system of a Discontinuous Galerkin type discretization. The third subject is the study of the stochastic pressure equation (without replacing the ordinary product by the Wick product). We use the white noise measure constructed from a Hilbert space and an operator to define and characterize adequate spaces for its solution. The approximation consists of a truncated Chaos expansion. We verify the well posedness of the discrete model and provide a priori error estimates. In all cases numerical experiments verify the theoretical results.