Finite Element Methods for High Order Eigenvalue Problems

特征值问题的数值计算在许多科学和工程应用中具有重要意义，如结构动力学、量子化学、电子网络、磁流体动力学、控制理论和逆问题。本项目主要研究由电磁逆散射理论引起的四旋度特征值问题（QCE）和麦克斯韦传输特征值问题（MTE）这两个高阶特征值问题的有效和高效的有限元方法。由于以下几个事实，MTE近年来受到了极大的关注：1)透射本征值与非散射波密切相关；2)透射特征值可以从散射数据中确定，从而在目标识别和无损检测中的各种反问题中发挥重要作用；3) MTE是非自伴随的，似乎不能用偏微分方程的标准技术来处理。提出的研究将为数学家和工程师提供可靠的QCE和MTE的新工具。此外，MTE的物理和理论还没有完全被理解。数值结果可能会把物理学家和数学家引向正确的方向。发展QCE和MTE的有限元方法有两个主要困难。特征值问题的计算通常从相应的源问题开始。高阶偏微分方程的有限元方法是一个具有挑战性的问题。第二个困难是对应源问题的成功方法可能不是特征值问题的好选择。为了克服上述困难，PI的目标是开发光谱正确的有限元方法，包括：1)QCE的不连续伽辽金法；2) MTE的迭代不连续Galerkin法；3) MTE二次重构的有限元方法。本研究将是高阶特征值问题有限元方法的重要进展。成功的结果将丰富有限元理论，特别是对于非自伴随和非线性特征值问题。

Numerical computation of eigenvalue problems is of fundamental importance in many scientific and engineering applications such as structural dynamics, quantum chemistry, electrical networks, magnetohydrodynamics, control theory, and inverse problems. This project focuses on developing effective and efficient finite element methods for two high order eigenvalue problems, the quad-curl eigenvalue problem (QCE) and the Maxwell's transmission eigenvalue problem (MTE), arising from the electromagnetic inverse scattering theory. The MTE has received significant attention recently due to several facts: 1) transmission eigenvalues are closely related to non-scattering waves; 2) transmission eigenvalues can be determined from scattering data and thus play an important role in a variety of inverse problems in target identification and nondestructive testing; 3) the MTE is non-selfadjoint and does not seem to be treatable by standard techniques for partial differential equations. The proposed research will provide mathematicians and engineers reliable new tools for the QCE and MTE. Furthermore, the physics and theory of the MTE are not yet fully understood. Numerical results may lead physicists and mathematicians in the correct direction.There are two major difficulties for developing finite element methods for the QCE and MTE. Computation of eigenvalue problems usually starts with the corresponding source problems. It is challenging to develop finite element methods for high order partial differential equations. The second difficulty is that a successful method for the corresponding source problem might not be a good choice for an eigenvalue problem. To overcome the above difficulties, the PI aims to develop spectrally correct finite element methods including 1) discontinuous Galerkin method for the QCE; 2) iterative discontinuous Galerkin method for the MTE; 3) finite element methods for a quadratic reformulation of the MTE. The proposed research will be an important advance of finite element methods for high order eigenvalue problems. Successful results will enrich finite element theories, in particular, for non-selfadjoint and nonlinear eigenvalue problems.