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

这一建议是对涉及旋度和div微分算子的偏微分方程的先进数值方法的研究，如麦克斯韦方程和线性弹性。研究的主题是多层自适应有限元方法的发展、应用和分析。该提案由三部分组成。第一部分重点研究了自适应有限元法在旋度和分度问题中的应用。提出了H（旋度）和H（div）问题的完整收敛理论，包括确定和不定麦克斯韦方程组和线性弹性的混合方法，并建立了一个框架来分析具有最小正则性假设的自适应网格上的局部多网格方法。第二部分是H（旋度）和H（分度）问题的算法发展。PI计划将牛顿迭代法和双网格法结合起来，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。另一个算法的发展是线性弹性问题的有限体积法和有限元法的耦合方法。第三部分是隐身装置的仿真。前两部分研究的理论和算法将用于模拟电磁波的近似隐身模型。PI建议使用AFEM，多重网格和高阶边缘元素开发一个软件包，可以为电磁材料的设计提供更多的见解。在这项工作中开发和研究的多层次自适应方法有望对大量实际问题的数值解决产生更广泛的影响。特殊的目标应用是麦克斯韦方程组和隐身装置的模拟。描述物质介质中电磁场演化的麦克斯韦方程在天线、微波、电路、电磁散射和无线技术的设计等方面有着广泛的实际应用。转换光学允许设计电磁材料，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、坦克或船只，从视图中消失。这种隐形衣很有前景，有一天可以作为防护罩，或者通过使信号阻塞的障碍物“消失”来改善无线通信。我们的数值模拟可以为这种新材料的设计提供见解。此外，对本科生和研究生计算数学教育的全面整合参与是该项目的一个组成部分。通过将代码纳入ifm软件包，PI将能够改进面向项目的适应性有限元方法课程，以便更好地教育和培训下一代计算数学家。

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.