Cluster-Robust Estimates for Galerkin and Petrov-Galerkin Discretizations of Elliptic Eigenvalue Problems

椭圆特征值问题的 Galerkin 和 Petrov-Galerkin 离散化的聚类鲁棒估计

• 批准号: 1522471
• 负责人: Jeffrey Ovall
• 金额: $ 14.99万
• 依托单位: Portland State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-15 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1522471&HistoricalAwards=false
• 关键词: Cluster; Robust; Estimates; Galerkin; Petrov

Eigenvalue problems for differential operators naturally arise in the study of vibrations in membranes and solids, fluid-solid interactions, and photonic crystals, and they also often play an important role in the practical analysis of many other time-dependent phenomena, such as acoustic or electromagnetic scattering. For many problems of interest, it is necessary to have provably efficient and robust means of estimating the error in computed approximations, as well as algorithms that can use this information to intelligently improve the approximations. This project concerns theoretical and algorithmic development of eigenvalue error estimates and self-adaptive methods for three computational approaches that promise to broaden the scope of available tools for addressing these challenging problems.The PI considers eigenvalue problems arising from second-order, linear, differential operators that are not necessarily self-adjoint, and which may have continuous components in their spectrum. The proposed work includes the development of a posteriori eigenvalue/eigenspace error estimates that are robust in the presence of repeated or tightly-clustered discrete eigenvalues---even if they are near components of the essential spectrum. Three broad classes of discretizations will be considered: penalty-based Discontinuous Galerkin (DG) methods, Discontinuous Petrov-Galerkin (DPG) methods, and so-called Implicit Element methods, which include variations on Virtual Element (VEM) methods and Boundary-Element-Based Finite Element (BEM-FEM) methods. In each case, the project will provide a posteriori error estimates that are cluster-robust in the sense that are insensitive to distances between true eigenvalues within the cluster that one is approximating, but instead depend on the relative distance between this cluster and the rest of the spectrum. In the case of DG methods, the PI expects to produce at least one provably-convergent, high-order adaptive method. In the case of implicit elements, he will first produce a high-order source-problem solver that is competitive with VEM and BEM-FEM, develop corresponding a posteriori error estimates, and then extend the approach to eigenvalue problems. In the case of DPG methods, he plans to exploit the fact that, with these techniques, indefinite source problems can be treated in a computational way that only involves self-adjoint and positive definite systems, and use this to derive a FEAST-like algorithm for computing large spectral clusters and/or clusters higher in the spectrum.

微分算子的本征值问题自然出现在膜和固体振动、流固相互作用和光子晶体的研究中，它们也经常在许多其他时间相关现象的实际分析中发挥重要作用，如声学或电磁散射。对于许多感兴趣的问题，需要有可证明的有效和鲁棒的方法来估计计算近似中的误差，以及可以使用这些信息来智能地改进近似的算法。本项目涉及三种计算方法的特征值误差估计和自适应方法的理论和算法发展，这些计算方法有望扩大解决这些具有挑战性问题的可用工具的范围。PI考虑由二阶、线性、微分算子引起的特征值问题，这些算子不一定是自伴随的，它们的谱中可能有连续的分量。提出的工作包括开发一种后验特征值/特征空间误差估计，这种估计在重复或紧密聚类的离散特征值存在时是鲁棒的——即使它们接近基本谱的分量。将考虑三大类离散化：基于惩罚的间断Galerkin （DG）方法，间断Petrov-Galerkin （DPG）方法和所谓的隐式单元方法，其中包括虚拟单元（VEM）方法和基于边界单元的有限元（BEM-FEM）方法的变体。在每种情况下，该项目将提供一个后验误差估计，该估计在某种意义上是集群鲁棒性的，对逼近的集群内真实特征值之间的距离不敏感，而是依赖于该集群与频谱其余部分之间的相对距离。在DG方法的情况下，PI期望产生至少一个可证明收敛的高阶自适应方法。在隐式元素的情况下，他将首先产生一个与VEM和BEM-FEM竞争的高阶源问题求解器，开发相应的后验误差估计，然后将该方法扩展到特征值问题。在DPG方法的案例中，他计划利用这样一个事实，即利用这些技术，不确定源问题可以用一种只涉及自伴随和正定系统的计算方式来处理，并利用这一点推导出一种类似feast的算法，用于计算大光谱簇和/或光谱中更高的簇。