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.