First-Order System Least Squares (FOSLS) for Partial Differential Equations

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AbstractThis is a renewal proposal to continue development of first--order system least squares (FOSLS) for numerical solution of partial differential equations (PDEs). It combines theoretical analysis, algorithm design, and software development, driven by several real applications, including aerodynamics, meteorology, elasticity, electromagnetics, particle transport, and porous flow. The goal is to develop accurate discretizations and fast solvers for the governing PDEs. The focus will be on the continued development of the FOSLS methodology, with special attention on developing methods that allow non-smooth problem character and solutions, and on further implementation of the methodology in the software package FOSPACK. Applications will include coupled systems, especially those arising in biological simulation, and porous media flow. Successful progress of this project would enable numerical simulations beyond current capabilities in many important applications of national interest.The central aim of this project is research in the field of computational mathematics. The purpose is to improve our understanding of the mathematics behind numerical 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, biological modeling, and material science. One of the challenges in such simulations is the development of improved computational methods for solving the mathematical 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 should 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 environment.