Least-Squares Finite Element Methods for Nonlinear Partial Differential Equations

本项目的目标是开发和分析流体和固体力学中应用的偏微分方程(PDE)的最小二乘方法。这些系统自然是非线性的，数值模拟通常是困难和昂贵的。最小二乘有限元方法是基于偏微分方程的许多科学和工程应用的有力工具。最小二乘法的主要特点之一是将给定的一组方程转化为一个松散耦合的标量方程组，便于用多层有限元方法求解。由适定最小二乘离散得到的线性方程组总是自伴且正定的，而变分多重网格法通常提供一个健壮的、可伸缩的解。此外，相关泛函本身提供了一个自然的局部误差估计器，可用于有效的自适应网格加密。将嵌套迭代和牛顿线性化与最小二乘离散化和多重网格迭代求解器相结合，构成了一种复杂的非线性问题的综合求解策略。本项目集中研究了几个线性和非线性弹性和流体流动问题的最小二乘方法，并将其扩展到粘弹性中的应用。这些领域的一些研究是成功的，初步结果令人鼓舞，但仍有许多工作要做。该项目既包括强非线性问题的一般求解方法，也包括尚未在最小二乘框架下分析的模型的具体最小二乘公式。该项目考虑的应用包括在工程、物理、空气动力学、大气科学、地质学和生物力学等领域非常重要的复杂和专业系统。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的非牛顿流体。这里，考虑悬浮红细胞的弹性性质的粘弹性模型是可以考虑的。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的效率的限制。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，加强更广泛的科学计算领域。

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.