The primary objective of this project is to develop a stochastic multiscale computational framework for modeling multiphysics systems arising in science and engineering applications with multiscale and uncertain input parameters. The mathematical models involve transient Stokes-Darcy systems coupled with systems of reaction-advection-diffusion equations. A multiblock domain decomposition methodology provides robust and efficient multiphysics and multinumerics couplings. The simulation domain is decomposed into a union of subdomains, each one associated with a physical, mathematical, and numerical model. Physically consistent interface conditions are imposed using mortar finite elements. Coarse scale mortar spaces lead to efficient and accurate multiscale approximations. Stochastic partial differential equations are employed to model uncertainty in the physical parameters. Sparse collocation methods are used for approximations in probability space. The project will investigate 1) mathematically rigorous and physically meaningful multiphysics models; 2) robust, accurate and efficient multiscale physical space and stochastic space discretization techniques; 3) a posteriori error estimates for adapting the models, the numerical grids in physical space, and the set of collocation points in stochastic space; 4) multiscale stochastic-based data assimilation and parameter estimation algorithms; 5) multiscale parallel domain decomposition solvers and preconditioners.

The work will emphasize computational modeling of energy and environment applications, in particular coupling of surface water with groundwater in hydrological systems, as well as biomedical applications such as modeling the inflammatory response in the human body. Both types of systems involve complex interactions of different physical processes and exhibit variability and uncertainty in the input parameters on a wide range of spatial and temporal scales. Computational modeling of coupled subsurface and surface flows and transport can provide reliable and cost effective predictions in contaminant remediation of rivers, lakes, wetlands, and aquifers. Inflammation plays a major role in the response of the human body to trauma, infection, or various diseases. Mathematical and computational modeling of these very complex processes can provide better understanding of their dynamics and spatial characteristics and may lead to the design of more effective treatments.

Project Report

The project focused on the mathematical modeling and computer simulation of science and engineering applications that involve interactions of multiple physical processes. Examples include interactions of surface water with groundwater in hydrological systems and their contaminant remediation, flow of oil and gas in fractured reservoirs, the inflammatory response in various diseases in the human body, and arterial flows. Taking into account complex multiphysics interactions in the computer models is critical for the accuracy of the simulations and the reliability of the predictions. The work on this project has lead to advances in modeling, discretizations, solution methods, and parameter estimation for complex multiphysics systems with applications to geoscience and biomedical problems. The research contributed in several ways to the computational practice of solving partial differential equations (PDEs). Since the solutions to most realistic PDE models cannot be found in analytical form, computer simulations play a crucial role in any modeling process. The goal is to compute a numerical solution which approximates the true solution. There are several important stages in this process to which this research has contributed: 1) develop and analyze mathematical models for multiphysics problems, 2) develop and analyze accurate discretization methods, 3) develop and analyze fast solution methods, 4) quantify uncertainty in the prediction via stochastic modeling, and 5) implement the methods in state-of-the-art computer simulators and apply them to real world science and engineering problems. In the first stage, mathematical models were developed based on partial differential equations allowing for different equations in different regions of the physical domain to account for the relevant physical processes. Examples include coupling the Stokes or the Navier-Stokes equations with Darcy flow. The flow systems were coupled with systems of reaction-advection-diffusion equations. Physically justified interface conditions were imposed on the interfaces between regions, such as conservation of mass and balance of forces. The existence and uniqueness of solutions to the resulting multuphysics variatonal formulations has been established. Contributions to the second stage include the development and analysis of new accurate and efficient locally mass conservative multiscale finite element methods suitable for rough grids and coefficients, as well as multiblock domains with possibly non-matching grids across interfaces and different mathematical and numerical models in different subdomains. The need for multiscale approximations comes from the wide range of variability in coefficients or parameters, e.g., permeability. The third stage requires solving very large systems of algebraic equations, often in the order of millions of unknowns, so it is crucial that these methods are very fast. Efficient multiscale domain decomposition solvers and preconditioners designed to scale well on massively parallel computers were developed. The work on the fourth stage includes development of stochastic methods combining adjoint based optimization, Kalman Filter, and Bayesian methods with stochastic collocation for computationally efficient data assimilation, parameter estimation, and uncertainty quantification in time dependent nonlinear PDEs. The methods developed in the first four stages have been implemented in state-of-the-art parallel computer simulators that model geoscience and biomedical applications.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Junping Wang
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University of Pittsburgh
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