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

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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.