Least-Squares Finite Element Methods for Nonlinear Partial Differential Equations

本项目的目标是开发和分析在流体和固体力学中应用的偏微分方程（PDEs）的最小二乘法。这些系统自然是非线性的，数值模拟通常是困难和昂贵的。最小二乘有限元法是许多基于pde的科学和工程应用的有力工具。最小二乘公式的主要特征之一是它将给定的一组方程转换成一个松散耦合的标量方程组，可以很容易地用多层有限元方法进行处理。单个未知数的有限元素空间可以根据简单性和可用性或潜在问题的物理特性独立选择。由适定最小二乘离散化得到的线性方程组总是自伴随的和正定的，变分多重网格方法通常提供了一个鲁棒的、可扩展的求解器。此外，相关函数本身提供了一个自然的锐局部误差估计，可用于有效的自适应网格细化。耦合嵌套迭代和牛顿线性化与近平方离散和多网格迭代求解器构成了求解非线性难题的一种强大、全面的求解策略。本项目集中研究了几种线性和非线性弹性和流体流动问题的最小二乘方法，并扩展到粘弹性的应用。这些领域的一些研究已经取得了成功，初步结果令人鼓舞，但仍有许多工作要做。该项目包括强非线性问题的一般解决方法和尚未在最小二乘框架中分析的模型的特定最小二乘公式。该项目考虑的应用包括工程、物理、空气动力学、大气科学、地质学和生物力学等领域的复杂和专业系统。例如，一个目标应用是在血流建模中出现的特定的不可压缩、非牛顿流。在这里，考虑到悬浮红细胞的弹性性质的粘弹性模型被考虑。在这些和许多其他感兴趣的领域，复杂现象的计算机模拟目前受到数值方法效率的限制。这里开发的基本技术将通过增加对如何分析和解决复杂问题的集体理解来增强科学计算的更广泛领域。

The goal of this project is the development and analysis of least-squaresmethods for partial differential equations (PDEs) arising from applicationsin fluid and solid mechanics. These systems are naturally nonlinear andnumerical simulation is typically difficult and expensive. Theleast-squares finite element method is a powerful tool for many PDE-basedapplications in science and engineering. One of the main characteristicsof a least-squares formulation is that it transforms a given set ofequations into a loosely coupled system of scalar equations that can betreated easily by multilevel finite element methods. The finite elementspaces for the individual unknowns may be chosen independently, based onsimplicity and availability or from the physics of the underlying problem.The linear systems of equations resulting from well-posed least-squaresdiscretizations are always self-adjoint and positive definite, andvariational multigrid methods generally provide a robust, scalable solver.In addition, the associated functional itself provides a natural sharplocal error estimator, which can be used for effective adaptive meshrefinement. Coupling nested iteration and Newton linearization with aleast-squares discretization and multigrid iterative solver constitutes arobust, comprehensive solution strategy for difficult nonlinear problems.This project represents a focused study of least-squares methods forseveral linear and nonlinear elasticity and fluid flow problems, extendingto applications in viscoelasticity. Some research in these areas has beensuccessful and preliminary results are encouraging, but much remains to bedone. This project includes both general solution methodologies forproblems with strong nonlinearities and specific least-squares formulationsfor models that have not been analyzed in a least-squares framework. Theapplications considered in this project include complex and specializedsystems important in areas including engineering, physics, aerodynamics,atmospheric sciences, geology, and biomechanics. One target application,for example, is a specific incompressible, non-Newtonain flow which arisesin the modeling of blood flow. Here, a viscoelastic model that takes intoaccount the elastic nature of the suspended red blood cells is to beconsidered. In these and many other areas of interest, computer simulationof complex phenomena are currently limited by efficiency of numericalmethods. The basic techniques developed here will enhance the broader areaof scientific computation by adding to the collective understanding of howto analyze and solve complex problems.