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

摘要本文是对偏微分方程组数值解的一阶系统最小二乘法(FOSLS)继续发展的更新建议。它结合了理论分析、算法设计和软件开发，由几个实际应用程序驱动，包括空气动力学、气象学、弹性学、电磁学、颗粒传输和渗流。其目标是为治理偏微分方程组开发精确的离散化和快速求解器。重点将放在继续发展FOSLS方法上，特别注意制定允许非平稳问题的性质和解决办法的方法，以及在FOSPACK软件包中进一步实施这一方法。应用将包括耦合系统，特别是那些在生物模拟中出现的系统，以及多孔介质流动。该项目的成功进展将使数值模拟能够在许多国家利益的重要应用中超越目前的能力。该项目的中心目标是计算数学领域的研究。其目的是提高我们对复杂物理现象的计算机数值模拟背后的数学知识的理解。这类模拟是研究和控制许多重要过程的关键，包括地下水流动、全球变化、能源生产、生物模拟和材料科学。这类模拟中的挑战之一是开发改进的计算方法来求解这些模型中出现的数学方程。这项研究的基本目标是大幅提高我们以更高的准确性和效率对日益复杂和复杂的过程进行建模的能力。这应该为模拟铺平道路，为科学家、工程师和政策制定者提供更强大的工具，以了解和改善我们的工业、科学和环境。

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.