Fast Algorithms for Models of Incompressible Flow

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FAST ALGORITHMS FOR MODELS OF INCOMPRESSIBLE FLOWHoward C. ElmanDepartment of Computer Science andInstitute for Advanced Computer StudiesUniversity of MarylandCollege Park, MD 20742This project concerns development, analysis and testing of efficient algorithms for solving systems of equations arising from models of flow of incompressible fluids. The development of such algorithms is of broad potential use for understanding scientific phenomena and constructing engineering tools involving fluid flows. Examples include biological flows such as blood flow in the heart or veins and arteries; dispersal of environmental pollutants in rivers or groundwater; design of microfluidics devices, which are used in diagnosis of diseases and identification of pathogens in fluids; and atmospheric and oceanic phenomena.Understanding such processes through purely experimental techniques is prohibitively expensive or impossible, whereas the use of modelling together with computational solution enables basic understanding of the physics by providing information about quantities such as flow rates, pressures, and concentrations of solvents. This research involves the development of fast computational algorithms that allows models to be resolved quickly and inexpensively by computer simulation.The research is focused on two classes of problems: algebraic eigenvalue problems that arise from analysis of the stability of solutions of the steady-state incompressible Navier-Stokes equations; and linear and nonlinear systems of equations that arise when thermal and chemical effects are coupled with models of incompressible flow. For both problem classes, discretization leads to the requirement to solve a sequence of large-scale linear systems of equations. We study efficient preconditioning techniques for these systems that take advantage of the structure of the problems. In addition to the impact on applied science, development of efficient solvers addresses fundamental mathematical and computational questions such as the impact of mathematical structure on algorithm development and the identification of stable flows in complex geometries.