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

微分算子的本征值问题自然出现在膜和固体，流体-固体相互作用和光子晶体中的振动研究中，并且它们在许多其他时间相关现象的实际分析中也经常发挥重要作用，例如声学或电磁散射。对于许多感兴趣的问题，有必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。这个项目涉及的理论和算法的发展特征值误差估计和自适应方法的三种计算方法，承诺扩大范围的可用工具来解决这些具有挑战性的problems.The PI认为特征值问题所产生的二阶，线性，微分算子不一定是自伴的，并可能有连续的成分，在其频谱。拟议的工作包括发展的后验特征值/特征空间误差估计是强大的存在下，重复或紧密聚集的离散特征值-即使他们是附近的基本频谱的组成部分。将考虑三大类离散化：基于罚函数的不连续伽辽金（DG）方法，不连续彼得罗夫-伽辽金（DPG）方法，和所谓的隐式单元方法，包括虚拟单元（VEM）方法和基于边界元的有限元（BEM-FEM）方法的变化。在每种情况下，该项目将提供一个后验误差估计，在这个意义上，它是集群鲁棒的，对近似的集群内的真实特征值之间的距离不敏感，而是取决于这个集群和频谱的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，他将首先产生一个高阶源问题求解器，是竞争与VEM和BEM-FEM，开发相应的后验误差估计，然后扩展到特征值问题的方法。在DPG方法的情况下，他计划利用这样一个事实，即利用这些技术，可以用只涉及自伴和正定系统的计算方式来处理不确定源问题，并使用它来推导一个类似FEAST的算法，用于计算大光谱簇和/或光谱中更高的簇。

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.