Least-Squares Finite Element Methods for Nonlinear Partial Differential Equations

这个项目的目标是发展和分析偏微分方程（PDE）的最小二乘方法，这些偏微分方程是在流体和固体力学中应用的。 这些系统是自然非线性和数值模拟是典型的困难和昂贵的。 最小二乘有限元法是科学和工程中许多基于偏微分方程的应用的有力工具。 最小二乘方法的一个主要特点是它将一组给定的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处理。 基于简单性和可用性，或者从基本问题的物理学出发，可以独立地选择各个未知量的有限元空间。由适定的最小二乘离散化产生的线性方程组总是自伴和正定的，并且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，相关的泛函本身提供了一个自然的尖锐局部误差估计器，其可用于有效的自适应网格细化。 将嵌套迭代和牛顿线性化与最小二乘离散和多重网格迭代求解器相结合，构成了解决复杂非线性问题的一种快速、全面的求解策略，本项目重点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，并将其应用扩展到粘弹性问题。 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。 这个项目包括两个一般的解决方法与强非线性问题和具体的最小二乘公式的模型，还没有被分析在最小二乘框架。 在这个项目中考虑的应用程序包括复杂和专业的系统，包括工程，物理，空气动力学，大气科学，地质学和生物力学领域的重要性。 一个目标应用，例如，是一个特定的不可压缩的，非牛顿流，这有助于血液流动的建模。在这里，考虑到悬浮的红细胞的弹性性质的粘弹性模型是被考虑。 在这些和许多其他领域的兴趣，计算机模拟的复杂现象，目前受到效率的数值方法。 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.