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

偏微分方程的特征值问题在科学和工程中有许多重要的应用，如清洁能源太阳能电池的设计、凝聚态物质电子结构的计算、超常光传输、无损检测、光子晶体和生物传感等。由于处理复杂结构的灵活性和严密的理论论证，有限元方法被广泛应用于计算特征值问题。线性特征值问题的有限元方法研究始于20世纪70年代，是一个活跃的研究领域。主要的泛函分析工具是线性算子的谱摄动理论。相反，对于非线性特征值问题，不存在系统的数值方法。有效的有限元方法是非常可取的。研究生和本科生都应该接受有关分析、建模和编程主题的培训。本项目的重点是发展新的非线性特征值问题的有限元方法。将这些问题转化为全纯Fredholm算子函数的特征值问题。利用抽象算子逼近理论分析了该算法的收敛性。将开发有效的数值方法来计算实际应用的特征值和/或特征向量。本文将研究两个重要的模型问题：色散光子晶体的能带结构计算和各向异性介质的透射特征值问题。由于非线性问题的特征值通常是复杂的，因此将开发在三维大问题的复平面上搜索特征值的高效算法。其新颖之处在于将全纯算子函数的谱理论与有限元逼近相结合。研究结果将推动有限元理论的发展，使科学家和工程师能够有效地计算非线性特征值问题。该项目还将提供一种新的方法来证明线性特征值问题的有限元方法的收敛性（包括一致性和非一致性）。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.