Numerical Modeling of Multiphysics Systems

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The main objectives of the proposed research are to design, analyze,and implement accurate, robust, and efficient numerical methods forpartial differential equations which model complex multiscalemultiphysics systems arising in science and engineeringapplications. In particular, the research will focus on severalcritical components of the mathematical and numerical modeling ofcoupled free fluid and porous media flows: identification of a correctmathematical model, especially appropriate interface conditions,analysis of the well-posedness of the model, design and analysis ofstable and accurate numerical discretization methods, including finiteelement methods on non-matching grids, interior convergence andpostprocessing, and a posteriori error estimation and adaptivity, anddevelopment of efficient solution algorithms such as parallel domaindecomposition and Newton-Krylov-multigrid solvers and preconditioners.The numerical techniques to be developed in this research are generaland address fundamental questions of numerical analysis of partialdifferential equations. Applications to coupling of groundwater andsurface water flows, as well as modeling of the cardiovascular systemwill be emphasized. Computer modeling of subsurface and surface flowand transport can provide reliable and cost-effective solutions toenvironmental and energy problems such as contaminant surface waterand groundwater remediation and enhanced oil recovery. Currentsimulators do not take into account the complex multiscale interactionbetween subsurface and surface geosystems. This research will modelthis interaction, which is crucial for the accurate prediction offluid behavior. The research will result in the development of robustand flexible software tools for performing accurate realisticmultiprocessor simulations of contaminant remediation of wetlands,lakes and estuaries. Mathematical and computational modeling of thecardiovascular system is a relatively new but rapidly growing area ofresearch. This application has a vast importance to society. Forexample, the heart attacks and strokes that result fromatherosclerosis are the major cause of death in industrialnations. The aim of the proposed research in this area is to modelcomputationally atherosclerosis taking into account the recentlydiscovered, but critical effect of inflammation. Coupled macroscopicmodels of blood flow within the artery and its wall will be combinedwith cellular level models of acute inflammatory response. Theresearch will lead to the development of optimized simulation toolswhich will advance prevention, detection, and therapy ofatherosclerosis.