Novel Finite Element Methods for Nonlinear Eigenvalue Problems - A Holomorphic Operator-Valued Function Approach

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Eigenvalue problems of partial differential equations have many important applications in science and engineering, e.g., design of solar cells for clean energy, calculation of electronic structure in condensed matter, extraordinary optical transmission, non-destructive testing, photonic crystals, and biological sensing. Due to the flexibility in treating complex structures and rigorous theoretical justification, finite element methods have been widely used to compute eigenvalue problems. The study of the finite element methods for linear eigenvalue problems started in the 1970s and has been an active research area since then. The main functional analysis tool is the spectral perturbation theory for linear operators. In contrast, for nonlinear eigenvalue problems, a systematic numerical approach does not exist. Effective finite element methods are highly desirable. Both graduate and undergraduate students are expected to receive training in the topics of analysis, modeling, and programming.This project focuses on the development of new finite element methods for nonlinear eigenvalue problems. These problems are recast as the eigenvalue problems of holomorphic Fredholm operator functions. The convergence will be analyzed using the abstract operator approximation theory. Effective numerical methods will be developed to compute the eigenvalues and/or eigenvectors for practical applications. Two important model problems will be studied: the band structure calculation of dispersive photonic crystals and the transmission eigenvalue problem for anisotropic media. Since the eigenvalues of nonlinear problems are complex in general, efficient algorithms to search eigenvalues on the complex plane for large problems in three dimensions will be developed. The novelty is the combination of the spectral theory for holomorphic operator functions and the finite element approximations. The results will advance the finite element theory and enable the scientists and engineers to effectively compute nonlinear eigenvalue problems. The project will also provide a new approach to prove the convergence of finite element methods (both conforming and non-conforming) for linear eigenvalue problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.