The transmission eigenvalue problem has attracted many researchers in the scattering and inverse scattering communities recently. Although simply stated, the problem is not covered by any standard theory of partial differential equations. Numerical treatment of transmission eigenvalues is very limited to date. Effective numerical methods will enhance the understanding of the problem and provide tools for mathematicians and engineers to compute transmission eigenvalues. This proposal aims at robust numerical methods for transmission eigenvalues for the Helmholtz equation and the Maxwell's equations. In particular, the following research topics will be carried out. 1) Iterative methods for the Helmholtz equation. Based on a fourth order reformulation, an associated generalized eigenvalue problem will be solved by the finite element method. Then iterative methods will be applied to search roots of a related algebraic function which turn out to be the transmission eigenvalues. 2) Continuous finite element method for the Maxwell's equations. The transmission eigenvalue problem of the Maxwell's equations will be written in a suitable weak form first. Then the curl conforming edge elements will be used to compute the transmission eigenvalues. 3) Iterative methods for the anisotropic Maxwell's equations. This approach is again based on a forth order reformulation of the transmission eigenvalue problem of the anisotropic Maxwell's equations. An associated generalized Maxwell's eigenvalue problem will be used to set up an algebraic equation whose roots are the transmission eigenvalues. Then iterative methods can be applied to search the roots of the algebraic equation. It is an extension of the iterative methods for the Helmholtz equation. However, the case for the Maxwell's equations is much more difficult and require additional technical treatment.
The proposed research will be a pioneer numerical study on transmission eigenvalues for the Helmholtz equation and the Maxwell's equations. The results are important for the development of mathematical theory for transmission eigenvalues and can be used to compare various estimates in inverse scattering theory. The proposed research will provide mathematicians and engineers reliable tools to compute transmission eigenvalues. It will lead to new methods for studying the inverse scattering problems such as inverse electromagnetic scattering problem for anisotropic media. Since transmission eigenvalues can be used to estimate material properties of the scattering object, the proposed research has potential usage in non-destructive testing, geophysical applications, medical imaging, etc. For example, it is possible to detect the presence of cavities in the dielectric from the location of the transmission eigenvalues. The numerical results will be disseminated to mathematician for analytical study of transmission eigenvalues and engineers for detection and reconstruction of unknown objects. In addition, successful accomplishment of the proposed project will enhance the research capacity of the university and provide graduate students valuable research opportunities.
