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

非线性特征值问题的新颖有限元方法 - 全纯算子值函数方法

• 批准号: 2109949
• 负责人: Jiguang Sun
• 金额: $ 10万
• 依托单位: Michigan Technological University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2109949&HistoricalAwards=false
• 关键词: Novel; Finite; Element; Methods; Nonlinear

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.

部分微分方程的特征值问题在科学和工程中具有许多重要的应用，例如，太阳能电池的设计用于清洁能源，凝结物质中电子结构的计算，非凡的光学传播，非破坏性测试，光子晶体以及生物学传感。由于在处理复杂结构和严格的理论理由方面具有灵活性，因此有限元方法已被广泛用于计算特征值问题。对线性特征值问题的有限元方法的研究始于1970年代，此后一直是一个活跃的研究领域。主要功能分析工具是线性操作员的光谱扰动理论。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。期望研究生和本科生都接受分析，建模和编程主题的培训。该项目重点介绍针对非线性特征值问题的新有限元方法的开发。这些问题是重新危险的，因为弗雷德·弗雷姆（Fredholm）操作员功能的特征值问题。将使用抽象操作员近似理论分析收敛性。将开发有效的数值方法来计算用于实际应用的特征值和/或特征向量。将研究两个重要的模型问题：分散光子晶体的带结构计算以及各向异性介质的传输特征值问题。由于非线性问题的特征值通常是复杂的，因此将开发出在复杂平面上搜索特征值以在三个维度中搜索大型问题的特征值。新颖性是骨膜操作员函数的光谱理论和有限元近似值的组合。结果将推进有限元理论，并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方法来证明线性特征值问题有限元方法（符合和不合格）的融合。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的审查标准来通过评估来获得支持的。