Numerical treatment of singularities and the Generalized Finite Element Method: theory, algorithms, and applications

奇点的数值处理和广义有限元方法：理论、算法和应用

• 批准号: 1016556
• 负责人: Victor Nistor
• 金额: $ 18万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-08-15 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1016556&HistoricalAwards=false
• 关键词: Numerical; treatment; singularities; Generalized; Finite

Numerical Methods for Partial Differential Equations (PDEs) are anessential ingredient in many areas of Sciences and Engineering. Oftenin the applications of PDEs, one has to deal with additionaldifficulties caused by the singularities in the geometry, thecoefficients, or the boundary conditions. The main goals of thisproposal are to obtain more efficient numerical methods for equationswith singular solutions and to develop user friendly implementations.The PI and his collaborators will design improved meshes that provideoptimal rates of convergence for the Finite Element Method in threedimensions for elliptic equations on polyhedral domains withnon-smooth interfaces. These equations form one of the basicingredients in many other applications. The results will be extendedto quasi-linear elliptic equations and to evolution equations. Forsome of these equations, spaces with higher smoothness or other additionalproperties are sometimes needed, and the Generalized Finite ElementMethod will be used for these purposes.The proposal will contribute to the formation of graduate andundergraduate students by advising and mentoring them and byintegrating them in research projects. The PI also plans to continueto run an Interdisciplinary Seminar featuring outside speakers,including non-mathematicians. The proposed research will lead to thedesign and testing of new and improved numerical methods for PartialDifferential Equations (PDEs) of interest in Sciences andEngineering. The resulting numerical methods will be presented andimplemented in a way that makes them accessible tonon-mathematicians. It will also contribute to applying mathematicalresults in practice by interaction with researchers from other fields,including the private sector. It will also contribute to the creationof specialists able to use advanced theoretical tools to handlepractical problems. He will also introduce numerical methods and theuse of computers in more traditional undergraduate courses. The PIalso plans to popularize mathematical questions that arise frompractice among mathematicians. The theoretical results and theresulting numerical methods will have applications in MathematicalPhysics, Quantum Chemistry, Biology, Finance (Risk Management andpricing of Financial Derivatives, or Options), and other areas.

偏微分方程 (PDE) 的数值方法是许多科学和工程领域的重要组成部分。在偏微分方程的应用中，人们常常必须处理由几何形状、系数或边界条件中的奇点引起的额外困难。该提案的主要目标是为具有奇异解的方程获得更有效的数值方法，并开发用户友好的实现。PI 和他的合作者将设计改进的网格，为具有非光滑界面的多面体域上的椭圆方程的三维有限元方法提供最佳收敛率。这些方程构成了许多其他应用的基本要素之一。结果将扩展到拟线性椭圆方程和演化方程。对于其中一些方程，有时需要具有更高平滑度的空间或其他附加属性，而广义有限元法将用于这些目的。该提案将通过为研究生和本科生提供建议和指导并将他们整合到研究项目中来促进研究生和本科生的培养。 PI 还计划继续举办由外部演讲者（包括非数学家）参加的跨学科研讨会。拟议的研究将导致科学和工程领域感兴趣的偏微分方程（PDE）的新的和改进的数值方法的设计和测试。由此产生的数值方法将以一种让非数学家也能理解的方式呈现和实现。它还将通过与其他领域（包括私营部门）的研究人员的互动，有助于将数学结果应用于实践。它还将有助于培养能够使用先进理论工具处理实际问题的专家。他还将在更传统的本科课程中介绍数值方法和计算机的使用。 PI 还计划向数学家普及实践中产生的数学问题。理论结果和由此产生的数值方法将应用于数学物理、量子化学、生物学、金融（金融衍生品或期权的风险管理和定价）和其他领域。