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.
