Gabriel Pinochet Soto | PhD in Mathematical Sciences Dissertation Defense

Title: Design and Implementation of Error Estimators for Finite Element Eigenvalue Problems

Abstract:

We present three publications, all encompassed under the umbrella of a posteriori error estimation theory for eigenvalue problems for finite element discretizations. The central objective of the research is the development of a general framework for the study of reliable estimation of eigenvalues and eigenspaces. We introduce applications to problems of theoretical interest as well as problems arising in real-life scenarios, such as optical fibers. The first paper focuses on the implementation of a dual-weighted residual error estimator for a nonselfadjoint eigenvalue problems arising from the study of leaky modes in optical fibers—Maxwell's equations, Perfectly Matched Layers, and a conforming finite element discretization are employed. Our method is eigensolver agnostic—we choose the FEAST algorithm for the computation of the eigenvalues. We display the utility of an h-adaptive algorithm driven by the estimator to capture fine-scale features of the eigenmodes. The second paper aims to explore the relationship between error estimation of the eigenvectors and eigenvalues, and error estimation arising from the associated landscape function problem. Selfadjoint elliptic eigenvalue problems are considered. Exploratory work is presented in a conforming Galerkin setting, which is further expanded when employing a symmetric interior penalty Discontinuous Galerkin (SIPG) discretization with hp-refinement. The third paper revisits the idea of rational approximations of the spectral projector (seen in the FEAST algorithm) and the source-problem based error estimation (seen in the landscape function approach) to provide a novel framework for eigenvalue error estimation. A generalized eigenspace decomposition for rational polynomials of nonselfadjoint operators is presented, based on the spectral mapping theorem. Further applications to gap error estimations are presented, in the context of a generic "source error estimator'' operator. We present Discontinuous Petrov-Galerkin and First-Order System Least Squares discretizations as particular instances of this framework, and showcase both selfadjoint and nonselfadjoint numerical examples, including microstructured optical fibers.