First-Order System Least Squares for Partial Differential Equations

偏微分方程的一阶系统最小二乘法

• 批准号: 9619792
• 负责人: Zhiqiang Cai
• 金额: $ 5万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-10-01 至 2000-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9619792&HistoricalAwards=false
• 关键词: First; Order; System; Least; Squares

9619792 Cai The investigator studies the numerical solution of partial differential equations (including convection-diffusion, Helmholtz, incompressible Stokes and Navier-Stokes, and elasticity equations) by a least-squares formulation for an equivalent first-order system. First-Order System Least Squares (FOSLS) formulates the original problem as an equivalent minimization problem by applying a least-squares principle to an equivalent first-order system. Hence, it represents a general methodology that can produce a variety of algorithms, depending on such choices as the first-order system and the least-squares norm, and that can lead to formulations that have substantially different numerical properties (accuracy, adaptivity, and complexity). The project seeks a proper first-order system and a proper least-squares norm for formulating the original problem so well that the numerical process (discretization and multigrid solution) becomes straightforward and optimal. Certain finite element methods for FOSLS minimization problems for second-order elliptic and incompressible Stokes and Navier-Stokes equations are of optimal accuracy in each variable (including new variables). Standard multigrid methods applied to the resulting discrete equations have optimal complexity (i.e., computational cost for full multigrid solution is proportional to the number of unknowns.). This project continues efforts on problems with discontinuous coefficients, high Reynolds number flows, and linear elasticity with general boundary conditions. This project aims to develop a new computational method for systems of parial differential equations that model the physical phenomena of fluid flow. The approach modifies existing mathematical formulations for these processes, and leads to more efficient and robust computational techniques than those currently in use. These new algorithms enable computer simulations of a large class of problems in science and engineering and mi nimize the need for costly experimental measurements.

小行星9619792 研究人员研究了偏微分方程（包括对流扩散，亥姆霍兹，不可压缩斯托克斯和Navier-Stokes方程和弹性方程）的数值解的最小二乘制定一个等效的一阶系统。 一阶系统最小二乘（FOSLS）通过将最小二乘原理应用于等效的一阶系统，将原始问题表示为等效的最小化问题。 因此，它代表了一种通用的方法，可以产生各种算法，这取决于一阶系统和最小二乘范数等选择，并可能导致配方，具有显着不同的数值属性（准确性，自适应性和复杂性）。 该项目寻求一个适当的一阶系统和一个适当的最小二乘规范制定原来的问题，使数值处理（离散化和多重网格解决方案）变得简单和最佳。 二阶椭圆不可压缩Stokes方程和Navier-Stokes方程的FOSLS极小化问题的某些有限元方法在每个变量（包括新变量）上都具有最佳精度。 应用于所得离散方程的标准多重网格方法具有最佳复杂性（即，完全多重网格解的计算成本与未知数的数量成比例。 本计画将继续研究不连续系数、高雷诺数流动、一般边界条件下的线弹性问题。 本计画旨在发展一种新的计算方法，以模拟流体流动的物理现象。 该方法修改了这些过程的现有数学公式，并导致比目前使用的更有效和更强大的计算技术。 这些新的算法使计算机模拟的一大类问题，在科学和工程，并最大限度地减少了昂贵的实验测量的需要。