This project focuses on multiscale and stochastic numerical methods for complex multiphysics problems. A parallel computational framework for modeling various multiscale applications will be developed. The models of interest will be based on coupling the Stokes or the Navier-Stokes equations with single phase or multiphase Darcy flow. The flow system will be coupled with reactive transport. The research approach is based on a multiblock domain decomposition methodology. The simulation domain is a union of subdomains (blocks). Each block is 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. Uncertainty in the physical characteristics is modeled via stochastic partial differential equations. Approximation in probability space is achieved using stochastic finite element and collocation methods. The main components of the research will be 1) development of multiscale mathematical models, identifying appropriate interface conditions, and analysis of the well-posedness of the models; 2) novel discretization techniques for stochastic partial differential equations; 3) a posteriori error estimates and adaptive mesh refinement algorithms in physical and stochastic space; and (4) efficient parallel domain decomposition solvers and preconditioners.
The research methodology can be applied for the design, analysis, and implementation of computational methods for a large class of problems arising in science and engineering applications. The project will emphasize accurate, robust, and efficient numerical methods for problems exhibiting behavior on a wide range of scales in space and time. The research also aims to quantify uncertainty in the prediction due to to uncertainty in the input parameters through stochastic modeling. The research results will be applied to two main application areas: couplings of surface water with groundwater flows and modeling complex multiscale processes occurring in the inflammatory response in the human body. Computer modeling of subsurface and surface flow and transport has a major economic impact on environmental and energy industries. Due to the proximity of groundwater and surface water systems, contamination of rivers, lakes, and wetlands can lead to contamination of aquifers and vice-versa. Simulating these complex interactions is a major computational challenge. Acute inflammation, which occurs in response to trauma or infection, may lead to a systemic inflammation or sepsis. Sepsis frequently leads to organ failure and death (with more that 500,000 fatal cases per year in the U.S.). Effective treatments are lacking, due to the complexity of the process. Mathematical and computer modeling is becoming a more common tool in the effort to gain insight of the dynamics and spatial characteristics of the inflammatory process.
