Multiscale domain decomposition methods for flow and mechanics problems

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A computational framework will be developed for modeling interactions of different physical phenomena. It will be applied to geoscience and biomedical problems of societal importance. Coupling subsurface and surface flow and transport will be investigated to model interactions between contaminated aquifers, rivers, lakes, and wetlands. Flows in fractured and deformable reservoirs will be modeled to provide improved understanding and predictive simulations of important processes occurring in hydraulic fracturing and carbon sequestration, including surface subsidence, pore collapse, cavity generation, and wellbore collapse. Another application of interest is flow in arteries, accounting for flow within the arterial wall. This has an effect on the blood velocity in the lumen and the speed of the pressure wave, as well as low density lipoproteins (LDL) transport and drugs filtered into the tissue during coronary artery flow. We expect the research on modeling arterial flows to lead to the development of optimized simulation tools which will advance drug delivery as well prevention, detection, and therapy of cardiovascular diseases. Educational activities will be integrated with and enhanced by research activities. Graduate students and postdocs will participate actively in research projects through research working groups or dissertation work. State-of-the-art research results will be incorporated into the curriculum. The primary objective of this work is to develop a computational framework for modeling multiphysics systems of coupled flow and mechanics problems with multiscale input parameters. The research approach is based on a multiblock domain decomposition methodology. The simulation domain is decomposed into a union of subdomains, each one associated with a physical, mathematical, and numerical model. Physically meaningful interface conditions are imposed on the discrete level via mortar finite elements. The formulation provides great flexibility for multiphysics and multinumerics couplings. Furthermore, this domain decomposition approach, combined with coarse scale mortar elements, provides a multiscale approximation and an efficient way to solve the coarse grid problem in parallel. The project will develop 1) Mathematically rigorous and physically meaningful multiphysics models; 2) Robust, accurate and efficient multiscale discretization techniques; 3) Efficient multiscale parallel domain decomposition solvers and preconditioners. The computational framework will be applied to geoscience and biomedical problems. We will develop variational formulations of systems of partial differential equations coupling free and porous media fluid flows with deformations of the porous solids. These formulations will couple through physically meaningful interface conditions free fluid models such as Stokes, Brinkman, or Navier-Stokes equations with single phase or multiphase Darcy flow. In regions involving deformable porous media the Darcy flow will be coupled with elasticity and modeled by the Biot system of poroelasticity. We will study well posedness of the variational formulations. We will develop stable and accurate multiscale mortar discretization methods for these multiphysics variational formulations. We will employ suitable mixed finite element, finite volume, and discontinuous Galerkin methods for the discretization of the subdomain equations on a fine scale. A mortar finite element space will be utilized to impose interface conditions on a coarse scale. We will carry out a priori multiscale error analysis for these methods. We will also develop efficient parallel non-overlapping domain decomposition algorithms for the solution of the resulting algebraic systems by reducing the coupled global multiscale problem to a coarse scale interface problem. We will analyze the condition number of the interface operator and will develop efficient preconditioners for speeding up the interface iteration.