Finite Element Methods for High Order Eigenvalue Problems

本征值问题的数值计算在结构动力学、量子化学、电网络、磁流体力学、控制理论和反问题等许多科学和工程应用中具有重要的意义。本项目致力于发展有效和高效的有限元方法来求解两个高阶特征值问题，即四旋度特征值问题(QCE)和麦克斯韦传输特征值问题(MTE)，这两个问题是由电磁逆散射理论产生的。由于以下几个事实，MTE最近受到了广泛的关注：1)传输本征值与非散射波密切相关；2)传输本征值可以从散射数据中确定，因此在目标识别和无损检测的各种逆问题中发挥着重要作用；3)MTE是非自伴的，似乎不能用标准的偏微分方程组方法来处理。所提出的研究将为数学家和工程师提供可靠的QCE和MTE新工具。此外，对MTE的物理和理论还没有完全了解。数值结果可能会将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。本征值问题的计算通常从相应的源问题开始。发展高阶偏微分方程组的有限元方法是一项具有挑战性的工作。第二个困难是，对于相应的源问题，一个成功的方法对于特征值问题可能不是一个好的选择。为了克服上述困难，PI致力于发展谱校正有限元方法，包括：1)QCE的间断Galerkin方法；2)MTE的迭代间断Galerkin方法；3)MTE的二次重构的有限元方法。本文的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.