First-Order System Least Squares for Partial Differential Equations

[919792]研究了一类等价一阶系统的偏微分方程（包括对流-扩散、Helmholtz、不可压缩Stokes和Navier-Stokes以及弹性方程）的最小二乘公式的数值解。一阶系统最小二乘（FOSLS）通过对等效一阶系统应用最小二乘原理，将原问题化为等效最小化问题。因此，它代表了一种可以产生各种算法的通用方法，这取决于一阶系统和最小二乘范数等选择，并且可以导致具有本质上不同数值特性（准确性、适应性和复杂性）的公式。该项目寻求一个适当的一阶系统和一个适当的最小二乘范数来很好地表述原始问题，使数值过程（离散化和多网格解）变得直接和最优。求解二阶椭圆型不可压缩Stokes方程和Navier-Stokes方程的FOSLS最小化问题的某些有限元方法在每个变量（包括新变量）上都具有最优精度。将标准多重网格方法应用于所得到的离散方程具有最佳的复杂性（即，完整多重网格解的计算成本与未知数的数量成正比）。本项目继续研究不连续系数、高雷诺数流动和一般边界条件下的线性弹性问题。本项目旨在为模拟流体流动物理现象的偏微分方程系统开发一种新的计算方法。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健壮的计算技术。这些新算法使计算机能够模拟科学和工程中的大量问题，并最大限度地减少对昂贵的实验测量的需要。

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.