Finite Element Methods for High Order Eigenvalue Problems

本征值问题的数值计算在许多科学和工程应用中具有根本的重要性，例如结构动力学、量子化学、电网络、磁流体力学、控制理论和逆问题。本计画主要针对电磁逆散射理论中两个高阶本征值问题，即四阶旋度本征值问题（QCE）与麦克斯韦传输本征值问题（MTE），发展有效且快速的有限元方法。由于透射特征值与非散射波密切相关，透射特征值可以由散射数据确定，在目标识别和无损检测中的各种反问题中起着重要作用，因此近年来MTE受到了广泛的关注。3）MTE是非自伴的，似乎不能用偏微分方程的标准技术来处理。该研究将为数学家和工程师提供可靠的QCE和MTE新工具。此外，MTE的物理和理论尚未完全理解。数值结果可以引导物理学家和数学家朝着正确的方向发展。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，一个成功的方法，相应的源问题可能不是一个很好的选择，一个特征值问题。为了克服上述困难，PI旨在开发谱校正有限元方法，包括1）用于QCE的间断Galerkin方法; 2）用于MTE的迭代间断Galerkin方法; 3）用于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一个重要进展。这些结果将丰富有限元理论，特别是非自伴和非线性特征值问题的有限元理论。

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.