Numerical Methods in Large-Scale Computation

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9706866 McCormick The main thrust of the proposed research is the development of efficient numerical techniques for simulation of a wide variety of physical processes. Important applications include aerodynamics, meteorology, elasticity, electromagnetics, porous media, and particle transport. The general goal is to develop accurate discretization methods and fast algebraic solvers for the partial differential equations that govern these and other applications. The research topics include: multilevel first-order system least squares, which involves reformulation of partial differential equations as well-posed minimization principles to allow for robust and efficient solution methods; porous media problems, which will be treated by Eulerian-Lagrangian localized adjoint methods that have been successful for such multiparty and reactive flows; transport phenomena, which will be simulated using efficient and robust least-squares methods for the three-dimensional Boltzmann transport equations; and iterative methods, which is aimed at developing effective algebraic solvers for the equations that arise in many applications. The focus of this project is research in the field of computational mathematics. The purpose is to improve our understanding of the mathematics behind computer simulation of complex physical phenomena. Such simulations are key to the study and control of many important processes, including groundwater flow, global change, energy production, and material science. One of the challenges in such simulations is the development of improved mathematical methods for solving the equations that arise in these models. The basic aim of this research is dramatic improvement in our ability to model increasingly more complicated and sophisticated processes with much greater accuracy and efficiency. This will pave the way for simulations that can provide scientists, engineers, and policy-makers with much more powerful tools to understand and improve our industry, science, and environm ent.