Theory, Algorithm and Appliction for H(curl) and H(div) Problems

H(curl)和H(div)问题的理论、算法和应用

• 批准号: 1115961
• 负责人: Long Chen
• 金额: $ 18万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-10-01 至 2014-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1115961&HistoricalAwards=false
• 关键词: Theory; Algorithm; Appliction; curl; div

This proposal is on the study of advanced numerical methods for partial differential equations that involve curl and div differential operators such as Maxwell's equations and linear elasticity. The theme of research is on the development, application, and analysis of multilevel adaptive finite element methods. The proposal consists of three parts. The first part focus on the theoretical investigation of Adaptive Finite Element Methods (AFEM) applied to H(curl) and H(div) problems. The PI propose to establish a complete convergence theory of AFEM for H(curl) and H(div) problems including definite and indefinite Maxwell's equations and mixed methods for linear elasticity, and to develop a framework to analyze local multigrid methods on adaptive grids with minimal regularity assumption. The second part is on the algorithmic development for H(curl) and H(div) problems. The PI plans to combine Newton's iteration and two-grid methods to develop a fast two-grid method for computing H(curl) and H(div) eigenvalue problems. Another algorithmic development is on a coupling method of finite volume method and finite element method for linear elasticity problems. The third part concentrates on the simulation of cloaking device. The theories and algorithms studied in the first two parts will be applied to simulate approximate cloaking models of electromagnetic waves. The PI propose to use AFEM, multigrid, and high order edge elements to develop a software package which could provide more insight on the design of electromagnetic materials.The multilevel adaptive methods developed and studied in this work are expected to have a broader impact on the numerical solutions of a large class of practical problems. Special target applications are Maxwell's equations and simulation of cloaking device.s Maxwell's equations describing the evolution of electromagnetic fields in material media have a wide range of practical applications such as design of antenna, microwave, circuits, electromagnetic scattering, and wireless technologies. The transformation optics allows for the design of electromagnetic materials that steer light around a hidden region, returning it to its original path on the far side. As a result, the contents of the hidden region, such as a helicopter, tank or ship, disappears from view. The cloaks show promise and could one day serve as protective shields or improve wireless communications by making signal-blocking obstacles "disappear." Our numerical simulation can provide insight on the design of such new materials. In addition, a fully integrated involvement in undergraduate and graduate computational mathematics education is an integral part of the project. By including code into the software package iFEM, the PI will be able to improve a project-oriented course on adaptive finite element methods for better education and training of the next generation of computational mathematicians.

该建议是针对涉及卷曲和Div差分运算符（例如Maxwell的方程式和线性弹性）的部分微分方程的高级数值方法的研究。研究的主题是关于多级自适应有限元方法的开发，应用和分析。 该提案包括三个部分。第一部分的重点是适用于H（Curl）和H（DIV）问题的自适应有限元方法（AFEM）的理论研究。 PI建议建立H（Curl）和H（DIV）问题的完整收敛理论，包括确定的和无限的Maxwell方程以及线性弹性的混合方法，并开发一个框架，以最小的规律性假设分析自适应网格的本地多机方法。第二部分是H（Curl）和H（DIV）问题的算法开发。 PI计划将牛顿的迭代和两网格方法结合起来，以开发一种快速的两网格方法来计算H（curl）和H（div）特征值问题。另一种算法开发是一种有限体积方法的耦合方法和有限元方法的线性弹性问题。第三部分集中于掩盖装置的模拟。在前两个部分中研究的理论和算法将用于模拟电磁波的近似掩盖模型。 PI建议使用AFEM，Multigrid和高级边缘元素来开发软件包，该软件包可以提供对电磁材料设计的更多见解。这项工作中开发和研究的多级自适应方法预计将对大量实用问题的数值解决方案产生更广泛的影响。特殊的目标应用是麦克斯韦的方程式和掩盖设备的模拟。SMAXWELL的方程描述了材料介质中电磁场演变的演变，具有广泛的实用应用，例如天线，微波，电路，电磁散射和无线技术。转换光学器件允许设计电磁材料，该材料在隐藏的区域周围引导灯光，并将其返回到远端的原始路径。结果，隐藏区域的内容（例如直升机，坦克或船舶）从视图中消失了。斗篷表现出希望，有一天可以用作保护性盾牌或通过使信号阻塞障碍“消失”来改善无线通信。我们的数值模拟可以提供有关此类新材料设计的见解。此外，全面综合参与了本科和研究生计算数学教育是该项目不可或缺的一部分。通过将代码纳入软件包IFEM中，PI将能够改善有关自适应有限元方法的面向项目的课程，以更好地教育和培训下一代计算数学家。