First-Order System Least Squares for Partial Differential Equations

9619792蔡研究了偏微分方程组(包括对流扩散方程、亥姆霍兹方程、不可压缩的Stokes方程、Navier-Stokes方程和弹性方程)的数值解。一阶系统最小二乘法(FOSLS)通过将最小二乘原理应用于等价的一阶系统，将原问题转化为等价的最小化问题。因此，它代表了一种通用的方法，它可以产生各种算法，取决于一阶系统和最小二乘范数等选择，并且可以导致具有显著不同的数值特性(精度、适应性和复杂性)的公式。该项目寻求一个合适的一阶系统和一个合适的最小二乘范数来表达原始问题，从而使数值过程(离散化和多重网格化)变得简单和最优。求解二阶椭圆型不可压缩Stokes方程和Navier-Stokes方程FOSLS极小化问题的某些有限元方法在每个变量(包括新变量)上都具有最优精度。应用于所得到的离散方程的标准多重网格法具有最优的复杂性(即，完全多重网格法的计算成本与未知数的数量成正比)。该项目继续研究具有不连续系数、高雷诺数流动和具有一般边界条件的线弹性问题。这个项目旨在开发一种新的计算方法，用于模拟流体流动的物理现象的偏微分方程组。该方法修改了这些过程的现有数学公式，并导致了比目前使用的更有效和更健壮的计算技术。这些新的算法能够对科学和工程中的一大类问题进行计算机模拟，并减少了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.