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

微分算符的本征值问题在膜和固体中的振动、流固相互作用和光子晶体的研究中自然产生，它们也经常在许多其他依赖时间的现象的实际分析中扮演重要角色，例如声学或电磁散射。对于许多感兴趣的问题，有必要具有可证明的高效和健壮的方法来估计计算近似中的误差，以及可以使用该信息来智能地改进近似的算法。这个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的发展，这些方法承诺扩大可用于解决这些具有挑战性的问题的工具的范围。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的，并且在它们的频谱中可能有连续的分量。建议的工作包括开发后验特征值/特征空间误差估计，该估计在存在重复或紧密聚集的离散特征值的情况下是稳健的-即使它们是基本谱的近分量。我们将考虑三大类离散：基于惩罚的间断Galerkin(DG)方法，间断Petrov-Galerkin(DPG)方法，以及所谓的隐式单元方法，包括基于虚拟单元的变形(VEM)方法和基于边界元的有限元(BEM-FEM)方法。在每种情况下，该项目都将提供一个后验误差估计，该估计在某种意义上是集群稳健的，它对正在逼近的集群内的真实本征值之间的距离不敏感，而是取决于该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。在隐式单元的情况下，他将首先产生一个与VEM和BEM-FEM相竞争的高阶源问题的求解器，并给出相应的后验误差估计，然后将该方法推广到本征值问题。在DPG方法的情况下，他计划利用这样一个事实，即使用这些技术，不确定源问题可以以一种只涉及自伴和正定系统的计算方式来处理，并利用这一点来推导出一种类似盛宴的算法来计算大型谱团和/或光谱中更高的团。

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.