A Newton-Galerkin Algorithm for Variational Investigations. Focus: Nonlinear Elliptic BVP

用于变分研究的牛顿-伽辽金算法。

• 批准号: 0074326
• 负责人: John Neuberger
• 金额: $ 8.5万
• 依托单位: Northern Arizona University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-09-01 至 2003-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0074326&HistoricalAwards=false
• 关键词: Newton; Galerkin; Algorithm; Variational; Investigations.

ABSTRACT:The proposed research primarily concerns finding, describing, and approximating solutions to semilinear elliptic zero-Dirichlet boundary value problems. While the Principal Investigator's first priority has been to prove existence and nodal structure theorems, his current efforts center on computational investigations. The PI will advance the numerical techniques used in the study of nonlinear differential equations in general and elliptic boundary value problems in particular. The PI's experiments support conjectures and reveal underlying structure as well as provide approximations. Secondary objectives include involving his students in the implementation of software solutions and refinement of algorithms. Lastly, the PI will prove convergence results relating to steepest descent, Newton's method, and Galerkin-type approximations. That these results may lead to analytical existence and nodal structure theorems is an additional source of motivation to the PI. The techniques to be used rely on the variational method, whereby solutions to the PDE are characterized as critical points of an associated nonlinear functional. The PI is currently in the process of decomposing function space into Newton flow-invariant manifolds and basins of attraction. These subsets of function space will be important to both numerical and analytical investigations.The proposed research is relevant to important areas of mathematics and science. Almost any scientific area concerns rates of change, hence differential equations. Many of these areas rely on the class of differential equations known as elliptic, and most of the difficult and physically significant problems are nonlinear. The PI's work is on the boundary of the computational and the analytical. An exciting new use of computational mathematics is in the investigation of nonlinear functional analysis. The PI has new techniques in nonlinear functional analysis for studying the underlying structure of this class of problem and believes that the techniques may generalize to an even wider class. Thus, funding this line of inquiry will benefit a large area of mathematics, soften or solve some long standing open problems in nonlinear elliptic differential equations, and be of use to many physical scientists.

摘要：本文主要研究半线性椭圆型零Dirichlet边值问题的求解、描述和近似。 虽然首席研究员的首要任务是证明存在性和节点结构定理，但他目前的努力集中在计算研究上。 PI将推进一般非线性微分方程和椭圆边值问题研究中使用的数值技术。 PI的实验支持结构，揭示潜在的结构，并提供近似值。 次要目标包括让他的学生参与软件解决方案的实现和算法的改进。 最后，PI将证明与最速下降、牛顿法和Galerkin型近似相关的收敛结果。 这些结果可能导致分析存在和节点结构定理是一个额外的动力来源PI。 要使用的技术依赖于变分法，从而解决方案的PDE的特征在于作为一个相关的非线性泛函的临界点。 PI目前正在将函数空间分解为牛顿流不变流形和吸引域。 这些函数空间的子集对数值和分析研究都很重要，所提出的研究与数学和科学的重要领域有关。 几乎所有的科学领域都关注变化率，因此微分方程。 这些领域中的许多都依赖于被称为椭圆型的微分方程，并且大多数困难和物理上重要的问题都是非线性的。 PI的工作是计算和分析的边界。 计算数学的一个令人兴奋的新用途是在非线性泛函分析的研究中。 PI在非线性泛函分析中有新的技术来研究这类问题的基本结构，并认为这些技术可以推广到更广泛的类别。 因此，资助这条调查线将有利于大面积的数学，软化或解决非线性椭圆微分方程中一些长期存在的开放问题，并对许多物理科学家有用。