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.

这项建议是关于研究偏微分方程组的高级数值方法，其中涉及卷曲和div算子，如麦克斯韦方程和线弹性。本文的研究主题是多层自适应有限元方法的发展、应用和分析。该提案由三个部分组成。第一部分对自适应有限元方法(AFEM)在H(Curl)和H(Div)问题中的应用进行了理论研究。PI建议建立H(Curl)和H(Div)问题的AFEM的完全收敛理论，包括定常和不定Maxwell方程以及线弹性混合方法，并在最小正则性假设下发展一个框架来分析自适应网格上的局部多重网格方法。第二部分是H(Curl)和H(Div)问题的算法发展。PI计划将牛顿迭代和双网格方法结合起来，开发一种计算H(旋度)和H(Div)特征值问题的快速双网格方法。另一个算法发展是线弹性问题的有限体积方法和有限元方法的耦合方法。第三部分对隐身装置进行了仿真研究。将前两部分研究的理论和算法应用于电磁波的近似隐身模型的仿真。PI建议使用AFEM、多重网格和高阶边缘单元来开发一个软件包，该软件包可以为电磁材料的设计提供更多的洞察力，本文开发和研究的多级自适应方法有望对一大类实际问题的数值解产生更广泛的影响。麦克斯韦方程描述了材料介质中电磁场的演化，具有广泛的实际应用，如天线设计、微波、电路、电磁散射和无线技术等。变换光学允许设计电磁材料来引导光线绕过隐藏区域，使其返回到远端的原始路径。因此，隐藏区域的内容(如直升机、坦克或舰船)将从视线中消失。这种斗篷显示出了希望，有朝一日可能会成为防护盾牌，或通过使信号阻挡障碍物“消失”来改善无线通信。我们的数值模拟可以为这些新材料的设计提供洞察。此外，全面参与本科生和研究生的计算数学教育是该项目的一个组成部分。通过将代码纳入IFEM软件包，PI将能够改进一门面向项目的自适应有限元方法课程，以更好地教育和培训下一代计算数学家。